%------------------------------------------------------------------------------
% File     : ITP268_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DeleteCorrectness 02043_129851
% 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    : 0073_VEBT_DeleteCorrectness_02043_129851 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 13773 (4557 unt;2748 typ;   0 def)
%            Number of atoms       : 26388 (9683 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 16895 (2435   ~; 527   |;1330   &)
%                                         (2706 <=>;9897  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of FOOLs       : 1469 ( 657 fml; 812 var)
%            Number of X terms     :  968 (   0  []; 770 ite; 198 let)
%            Number of types       :  559 ( 558 usr)
%            Number of type conns  : 2515 (1701   >; 814   *;   0   +;   0  <<)
%            Number of predicates  :  151 ( 148 usr;   2 prp; 0-4 aty)
%            Number of functors    : 2065 (2065 usr; 512 con; 0-7 aty)
%            Number of variables   : 27129 (26396   !; 733   ?;27129   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 11:22:12.673
%------------------------------------------------------------------------------
% Could-be-implicit typings (558)
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tff(ty_n_t__fun_It__Set__Oset_It__Complex__Ocomplex_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__fun_It__Rat__Orat_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
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tff(ty_n_t__fun_It__Option__Ooption_It__Num__Onum_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Option__Ooption_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_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__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Real__Oreal_J_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__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Rat__Orat_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__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(ty_n_t__fun_It__String__Ochar_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Ochar_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Rat__Orat_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__Nat__Onat_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__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_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__Rat__Orat_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__List__Olist_I_Eo_J_M_Eo_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__Filter__Ofilter_It__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_I_Eo_J_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__Option__Ooption_It__Nat__Onat_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__Set__Oset_It__String__Ochar_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__fun_I_Eo_Mt__String__Ochar_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__Num__Onum_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__List__Olist_It__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_M_Eo_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__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__fun_It__Rat__Orat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Num__Onum_M_Eo_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
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tff(ty_n_t__Extended____Nat__Oenat,type,
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tff(ty_n_t__List__Olist_I_Eo_J,type,
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tff(ty_n_t__Complex__Ocomplex,type,
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tff(ty_n_t__Set__Oset_I_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
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tff(ty_n_t__String__Ochar,type,
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tff(ty_n_t__Real__Oreal,type,
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tff(ty_n_t__Rat__Orat,type,
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tff(ty_n_t__Num__Onum,type,
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tff(ty_n_t__Nat__Onat,type,
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tff(ty_n_t__Int__Oint,type,
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% Explicit typings (2190)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: complex > fun_na1554462459066888140plex_o ).

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,
    aTP_Lamm_aad: fun_real_fun_real_o ).

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__aai____,type,
    aTP_Lamm_aai: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__aaj____,type,
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tff(sy_c_ATP_058Lamm__aak____,type,
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tff(sy_c_ATP_058Lamm__aal____,type,
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tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__aan____,type,
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tff(sy_c_ATP_058Lamm__aao____,type,
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tff(sy_c_ATP_058Lamm__aap____,type,
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tff(sy_c_ATP_058Lamm__aaq____,type,
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tff(sy_c_ATP_058Lamm__aar____,type,
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tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: int > fun_in2204242111582549184nt_int ).

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_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_nat_complex > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__abc____,type,
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tff(sy_c_ATP_058Lamm__abd____,type,
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tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__abh____,type,
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tff(sy_c_ATP_058Lamm__abi____,type,
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tff(sy_c_ATP_058Lamm__abj____,type,
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tff(sy_c_ATP_058Lamm__abk____,type,
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tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__abn____,type,
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tff(sy_c_ATP_058Lamm__abo____,type,
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tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__abr____,type,
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tff(sy_c_ATP_058Lamm__abs____,type,
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tff(sy_c_ATP_058Lamm__abt____,type,
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tff(sy_c_ATP_058Lamm__abu____,type,
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tff(sy_c_ATP_058Lamm__abv____,type,
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tff(sy_c_ATP_058Lamm__abw____,type,
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tff(sy_c_ATP_058Lamm__abx____,type,
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tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: fun_na1668100129766719005at_nat ).

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

tff(sy_c_ATP_058Lamm__aca____,type,
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tff(sy_c_ATP_058Lamm__acb____,type,
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tff(sy_c_ATP_058Lamm__acc____,type,
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tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: fun_int_nat ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__acf____,type,
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tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: $o > fun_o_5488095450495718970o_char ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: $o > fun_o_6360995121869435791o_char ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: $o > fun_o_2683029531876921444o_char ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: $o > fun_o_3037882179896959673o_char ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: $o > fun_o_335877185394996366o_char ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: $o > fun_o_2016307092959984611o_char ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: $o > fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_nat_real > fun_real_real ).

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

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: fun_real_real ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_real_real ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: fun_nat_real ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_nat_real ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_nat_real ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: fun_real_real ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__afi____,type,
    aTP_Lamm_afi: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: real > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__afm____,type,
    aTP_Lamm_afm: set_complex > fun_co51012501629074457plex_o ).

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: set_real > fun_re6988797352886916895real_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_real_real ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_int ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_complex_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_int_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_real_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_Pr958786334691620121nt_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_real_real ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_int_int ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_nat_o > fun_fu4001317426178549763_nat_o ).

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

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

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

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_complex_o > fun_fu1358370060594899587plex_o ).

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

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_int > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_complex_complex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_co7673397065115134084omplex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_complex_complex > fun_co7673397065115134084omplex ).

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

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: nat > fun_fu7002938529518989751at_nat ).

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

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: set_int > fun_fu7328815047503086531_int_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: set_int > fun_fu8404755328348021045al_int ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: set_real > fun_fu6324116013759063093nt_int ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_int > fun_fu810243669753863363_int_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_int > fun_fu8001260946965626035ex_int ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_complex > fun_fu2339071201290874307plex_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: set_complex > fun_fu7424888833511495217nt_int ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: set_nat > fun_fu5253574275612187245l_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_real > fun_fu5321668254738230545t_real ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_int_int > fun_int_fun_int_int ).

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

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

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

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

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

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_int_complex > fun_int_o ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_complex_nat > fun_complex_o ).

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

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_nat_complex > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_rat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_na4138050610385705875at_nat ).

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

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

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_complex > fun_nat_complex ).

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

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

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_Pr3238374608701176162_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_complex_o > fun_complex_o ).

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

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

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

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

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_nat > fun_fu7002938529518989751at_nat ).

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

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

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_nat_real > fun_na7718645245721260691t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_nat_rat > fun_fu8328862064278946247at_rat ).

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

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_int > fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_nat_int > fun_nat_int ).

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

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

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

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

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

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

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_nat > fun_nat_Code_integer ).

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

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

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_int ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_nat_int > fun_fu8464300013002972271at_int ).

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

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

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_co8378349014452122532t_real > fun_na5723442673449079972x_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: set_complex > fun_fu7426956082801719578t_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: set_real > fun_fu5104398059618711576t_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: set_int > fun_fu7141299348810748056t_real ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_re2084383175013312036omplex > fun_na1957955181367619364omplex ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: set_real > fun_fu847012557204443036omplex ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_na1142814060743658952omplex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: set_nat > fun_fu7808106308057563584omplex ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_in8056368165505483812omplex > fun_na5558608354090810916omplex ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: set_int > fun_fu6465309499045537820omplex ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: set_int > fun_fu688336635129761048at_int ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_co3258106519541737638omplex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: set_complex > fun_fu689363623238371998omplex ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: set_nat > fun_fu7979797992566127108at_nat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: set_nat > fun_fu2115079063411676732t_real ).

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

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: $o > fun_o_o ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_co8378349014452122532t_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_int_fun_nat_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_re2084383175013312036omplex > fun_real_complex ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_na1142814060743658952omplex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_in8056368165505483812omplex > fun_int_complex ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_int_fun_nat_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_co3258106519541737638omplex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_o_nat > fun_fu6701067901881939075_o_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_nat_Code_integer > fun_Co6226381647958292740nteger ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: int > fun_nat_fun_nat_int ).

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

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

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

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

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_nat_complex > fun_na1591161823130732966omplex ).

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

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

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_nat_complex > fun_se8708249003167407990omplex ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_real > fun_se2100927751953696370t_real ).

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

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_real > fun_re4238438383274954274t_real ).

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: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_na1142814060743658952omplex ).

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

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

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

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_na1142814060743658952omplex ).

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

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

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

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_fun_nat_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_nat ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_nat_fun_nat_nat ).

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

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

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_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_nat_complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_real > fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_int > fun_int_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: real > fun_re4756235295228751087t_real ).

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

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: nat > fun_fu1620311477532340197t_real ).

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_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_na1142814060743658952omplex ).

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

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

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

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_na1142814060743658952omplex ).

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

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

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

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_int_int ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: fun_real_real ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: code_integer > fun_Co5080488584548667985nteger ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_na1142814060743658952omplex ).

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

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: fun_nat_int ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_int_int ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: fun_nat_real > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: int > fun_int_fun_int_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__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_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: fun_co3258106519541737638omplex ).

tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: fun_int_fun_nat_int ).

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

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

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: fun_num_fun_num_num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun_Pr4799464687642144302_num_o ).

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__Int__Oint,type,
    bit_se725231765392027082nd_int: fun_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,
    bit_se1345352211410354436nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: fun_nat_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: fun_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_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

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

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

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_Ointeger__of__num,type,
    code_integer_of_num: fun_num_Code_integer ).

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

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun_num_Code_integer ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Numeral_Opositive,type,
    code_positive: fun_num_Code_integer ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun_num_int ).

tff(sy_c_Code__Target__Int_Opositive,type,
    code_Target_positive: fun_num_int ).

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__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
    complete_Sup_Sup_int: set_int > int ).

tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

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_Ocnj,type,
    cnj: fun_complex_complex ).

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

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: fun_complex_real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: fun_complex_real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: fun_complex_complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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_Oadjust__div,type,
    adjust_div: fun_Pr982841511892225912nt_int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: fun_int_fun_int_int ).

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

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int ) > fun_Pr3238374608701176162_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: fun_Pr3238374608701176162_int_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: fun_Pr4796943543331071514_nat_o ).

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tff(sy_c_List_Oproduct_001_Eo_001_Eo,type,
    product_o_o: ( list_o * list_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: ( list_nat * list_o ) > list_P7333126701944960589_nat_o ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
    product_nat_int: ( list_nat * list_int ) > list_P3521021558325789923at_int ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: ( list_nat * list_nat ) > list_P6011104703257516679at_nat ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
    product_nat_num: ( list_nat * list_num ) > list_P1726324292696863441at_num ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
    product_num_num: ( list_num * list_num ) > list_P3744719386663036955um_num ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oreplicate_001_Eo,type,
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tff(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
    replicate_complex: ( nat * complex ) > list_complex ).

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,
    replicate_real: ( nat * real ) > list_real ).

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

tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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

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

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

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,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__Extended____Nat__Oenat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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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__Code____Numeral__Ointeger,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat__Bijection_Olist__encode,type,
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tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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tff(sy_c_Nat__Bijection_Oprod__encode,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
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tff(sy_c_NthRoot_Osqrt,type,
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tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Num_Onum_Osize__num,type,
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tff(sy_c_Num_Onum__of__nat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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tff(sy_c_Num_Opow,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Num_Osqr,type,
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tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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tff(sy_c_Power_Opower__class_Opower_001t__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__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,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_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__String__Ochar,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__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_M_Eo_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Rat_OFract,type,
    fract: int > fun_int_rat ).

tff(sy_c_Rat_OFrct,type,
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tff(sy_c_Rat_Onormalize,type,
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tff(sy_c_Rat_Oquotient__of,type,
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tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
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tff(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,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__Real__Oreal,type,
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tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,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__Rat__Orat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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    suminf_real: fun_nat_real > real ).

tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > $o ).

tff(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: fun_nat_int > $o ).

tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: fun_nat_nat > $o ).

tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_o ).

tff(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: fun_nat_int > fun_int_o ).

tff(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: fun_nat_nat > fun_nat_o ).

tff(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: fun_nat_real > fun_real_o ).

tff(sy_c_Set_OCollect_001_Eo,type,
    collect_o: fun_o_o > set_o ).

tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: fun_Code_integer_o > set_Code_integer ).

tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: fun_complex_o > set_complex ).

tff(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: fun_int_o > set_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
    collect_list_o: fun_list_o_o > set_list_o ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: fun_list_complex_o > set_list_complex ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: fun_list_int_o > set_list_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_o > set_list_nat ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: fun_list_VEBT_VEBT_o > set_list_VEBT_VEBT ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_o > set_nat ).

tff(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: fun_num_o > set_num ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    collec8663557070575231912omplex: fun_Pr7716844367077540446plex_o > set_Pr5085853215250843933omplex ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: fun_Pr3238374608701176162_int_o > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: fun_Pr4796943543331071514_nat_o > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    collec3799799289383736868l_real: fun_Pr4814023175418636514real_o > set_Pr6218003697084177305l_real ).

tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: fun_rat_o > set_rat ).

tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: fun_real_o > set_real ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: fun_set_complex_o > set_set_complex ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: fun_set_int_o > set_set_int ).

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

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

tff(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
    image_o_set_o: ( fun_o_set_o * set_o ) > set_set_o ).

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__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    image_int_set_int: ( fun_int_set_int * set_int ) > set_set_int ).

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__Real__Oreal,type,
    image_nat_real: ( fun_nat_real * set_nat ) > set_real ).

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__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

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_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

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

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

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
    set_fo1084959871951514735nteger: ( fun_na7078387940644931716nteger * nat * nat * code_integer ) > code_integer ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat ) > fun_na1591161823130732966omplex ).

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__Num__Onum,type,
    set_fo8365102181078989356at_num: ( fun_nat_fun_num_num * nat ) > fun_nat_fun_num_num ).

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_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
    set_fo3699595496184130361el_nat: fun_Pr8463018485500633518_nat_o ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Num__Onum,type,
    set_fo256927282339908995el_num: fun_Pr7588637180273442094_num_o ).

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_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

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

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__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_001_Eo,type,
    set_or6416164934427428222Than_o: fun_o_set_o ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
    set_or1207661135979820486an_int: fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
    set_ord_lessThan_o: fun_o_set_o ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: fun_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_String_Oascii__of,type,
    ascii_of: char > char ).

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

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

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

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Topological__Spaces_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_Ogenerate__topology_001_Eo,type,
    topolo4667128019001906403logy_o: set_set_o > fun_set_o_o ).

tff(sy_c_Topological__Spaces_Ogenerate__topology_001t__Int__Oint,type,
    topolo1611008123915946401gy_int: set_set_int > fun_set_int_o ).

tff(sy_c_Topological__Spaces_Ogenerate__topology_001t__Nat__Onat,type,
    topolo1613498594424996677gy_nat: set_set_nat > fun_set_nat_o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Code____Numeral__Ointeger,type,
    topolo2919662092509805066nteger: fun_nat_Code_integer > $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_Oopen__class_Oopen_001_Eo,type,
    topolo9180104560040979295open_o: fun_set_o_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
    topolo4110288021797289639omplex: set_complex > $o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Int__Oint,type,
    topolo4325760605701065253en_int: fun_set_int_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Nat__Onat,type,
    topolo4328251076210115529en_nat: fun_set_nat_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
    topolo4860482606490270245n_real: set_real > $o ).

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

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

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

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

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
    diffs_Code_integer: fun_nat_Code_integer > fun_nat_Code_integer ).

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

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

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

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

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

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

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: fun_Pr1283576600272279410_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
    vEBT_V452583751252753300el_num: fun_Pr4822914654802553862_num_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V7235779383477046023at_nat: fun_Pr4779616527126863150_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_o ).

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

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

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

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
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tff(sy_c_Wellfounded_Omeasure_001t__Num__Onum,type,
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tff(sy_c_Wellfounded_Opred__nat,type,
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tff(sy_c_aa_001_Eo_001_Eo,type,
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    aa_fun789424914597169340plex_o: ( fun_fu3043630359712370243plex_o * fun_re7319081385389557872plex_o ) > fun_re7319081385389557872plex_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_M_Eo_J_J_Mt__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    aa_fun6739043078600750748omplex: ( fun_fu1853463631764337203omplex * fun_re7208826350674946562omplex ) > fun_fu3572752990780465756omplex ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Complex__Ocomplex_M_Eo_J_J_Mt__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J_J,type,
    aa_fun7501666206714588442omplex: ( fun_fu7433039812794084273omplex * fun_re7208826350674946562omplex ) > fun_fu4045816715887884506omplex ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_fun3996843141751676939nt_int: ( fun_fu7572495796013443162nt_int * fun_real_fun_int_o ) > fun_int_int ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__Real__Oreal_M_Eo_J_J,type,
    aa_fun2391495421969208892real_o: ( fun_fu8602227297639178947real_o * fun_real_fun_int_o ) > fun_int_fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
    aa_fun7555737461897725707al_int: ( fun_fu86166166672686426al_int * fun_real_fun_int_o ) > fun_real_int ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    aa_fun2967211649596192572_int_o: ( fun_fu7328815047503086531_int_o * fun_real_fun_int_o ) > fun_real_fun_int_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J,type,
    aa_fun129049606478405534nt_int: ( fun_fu6324116013759063093nt_int * fun_real_fun_int_int ) > fun_fu7572495796013443162nt_int ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Int__Oint_M_Eo_J_J_Mt__fun_It__Real__Oreal_Mt__Int__Oint_J_J,type,
    aa_fun1679075639499870878al_int: ( fun_fu8404755328348021045al_int * fun_real_fun_int_int ) > fun_fu86166166672686426al_int ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun7366046364592083119at_nat: ( fun_fu8066363281484412158at_nat * fun_real_fun_nat_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun6247686994378451851t_real: ( fun_fu8372050067615010778t_real * fun_real_fun_nat_o ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_M_Eo_J_J,type,
    aa_fun6917832400337718588real_o: ( fun_fu4757868175734369219real_o * fun_real_fun_nat_o ) > fun_nat_fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
    aa_fun2272639302901571595al_nat: ( fun_fu2338672169953838170al_nat * fun_real_fun_nat_o ) > fun_real_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun6320755270648597735l_real: ( fun_fu4971871526818099510l_real * fun_real_fun_nat_o ) > fun_real_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun3244926128379188796_nat_o: ( fun_fu5289434066392675523_nat_o * fun_real_fun_nat_o ) > fun_real_fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    aa_fun7632768522510439005omplex: ( fun_fu847012557204443036omplex * fun_re2084383175013312036omplex ) > fun_nat_complex ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun7718511627822915834at_nat: ( fun_fu5553606195750802321at_nat * fun_real_fun_nat_nat ) > fun_fu8066363281484412158at_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    aa_fun7039802393428226646al_nat: ( fun_fu8695033588447641837al_nat * fun_real_fun_nat_nat ) > fun_fu2338672169953838170al_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun628238740774635481t_real: ( fun_fu5104398059618711576t_real * fun_re4238438383274954274t_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    aa_fun7274880274332377722t_real: ( fun_fu5321668254738230545t_real * fun_re4238438383274954274t_real ) > fun_fu8372050067615010778t_real ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    aa_fun6240130701872165846l_real: ( fun_fu5253574275612187245l_real * fun_re4238438383274954274t_real ) > fun_fu4971871526818099510l_real ).

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

tff(sy_c_aa_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    aa_fun3386010016858254564at_nat: ( fun_fu1590586450949517859at_nat * fun_set_nat_nat ) > fun_set_nat_nat ).

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

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

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: ( complex * set_complex ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: ( int * set_int ) > $o ).

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

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

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

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

tff(sy_c_member_001t__Num__Onum,type,
    member_num: ( num * set_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    member1379723562493234055eger_o: ( produc6271795597528267376eger_o * set_Pr448751882837621926eger_o ) > $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__Nat__Onat_Mt__Num__Onum_J,type,
    member9148766508732265716at_num: ( product_prod_nat_num * set_Pr6200539531224447659at_num ) > $o ).

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

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: ( real * set_real ) > $o ).

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

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

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

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_lx____,type,
    lx: 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_summin____,type,
    summin: nat ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (11023)
tff(fact_0__C1_C,axiom,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

% "1"
tff(fact_1_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_2_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_3__C3_C,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "3"
tff(fact_4__C5_OIH_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(vEBT_invar_vebt(X),na)
        & ! [Xa: nat] : aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(X,Xa)),na) ) ) ).

% "5.IH"(1)
tff(fact_5_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_6_bit__split__inv,axiom,
    ! [Xb: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).

% bit_split_inv
tff(fact_7_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_delete(Ta,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_8__C5_Ohyps_C_I8_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.hyps"(8)
tff(fact_9__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_10_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_11_high__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_12__092_060open_062invar__vebt_A_ItreeList_A_B_Asummin_J_An_092_060close_062,axiom,
    aa_nat_o(vEBT_invar_vebt(nth_VEBT_VEBT(treeList,summin)),na) ).

% \<open>invar_vebt (treeList ! summin) n\<close>
tff(fact_13__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_14__C5_OIH_C_I2_J,axiom,
    ! [Xb: nat] : aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(summary,Xb)),m) ).

% "5.IH"(2)
tff(fact_15__C0_C,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
     => aa_nat_o(vEBT_invar_vebt(X),na) ) ).

% "0"
tff(fact_16__C12_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg) ).

% "12"
tff(fact_17__092_060open_062_092_060exists_062z_O_Aboth__member__options_A_ItreeList_A_B_Asummin_J_Az_092_060close_062,axiom,
    ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,summin)),X_1) ).

% \<open>\<exists>z. both_member_options (treeList ! summin) z\<close>
tff(fact_18_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_19_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_20__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na)) ).

% \<open>both_member_options summary (high ma n)\<close>
tff(fact_21_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia != Maa )
       => ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_22_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Y ) ) ).

% high_inv
tff(fact_23_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Xb ) ) ).

% low_inv
tff(fact_24__C8_C,axiom,
    aa_nat_nat(suc,na) = m ).

% "8"
tff(fact_25__C4_C,axiom,
    ! [I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),X_12)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I) ) ) ).

% "4"
tff(fact_26__092_060open_062both__member__options_A_ItreeList_A_B_Ahigh_Ama_An_J_A_Ilow_Ama_An_J_092_060close_062,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) ).

% \<open>both_member_options (treeList ! high ma n) (low ma n)\<close>
tff(fact_27_nnvalid,axiom,
    aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),na) ).

% nnvalid
tff(fact_28_hprolist,axiom,
    nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)) = vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)) ).

% hprolist
tff(fact_29__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% \<open>summin * 2 ^ n + lx < 2 ^ deg\<close>
tff(fact_30_newsummvalid,axiom,
    aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),m) ).

% newsummvalid
tff(fact_31__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_092_060close_062,axiom,
    size_s6755466524823107622T_VEBT(treeList) = size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))) ).

% \<open>length treeList = length (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)])\<close>
tff(fact_32_allvalidinlist,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))))
     => aa_nat_o(vEBT_invar_vebt(X),na) ) ).

% allvalidinlist
tff(fact_33_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) )
          & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_34_ninNullc,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))) ).

% ninNullc
tff(fact_35_delt__out__of__range,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% delt_out_of_range
tff(fact_36__092_060open_062summin_A_060_A2_A_094_Am_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)) ).

% \<open>summin < 2 ^ m\<close>
tff(fact_37_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_38__C118_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),
      $ite(
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
        $let(
          maxs: option_nat,
          maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
          $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
        ma )) ).

% "118"
tff(fact_39__C117_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,
        $ite(
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
          $let(
            maxs: option_nat,
            maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
            $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
          ma )),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "117"
tff(fact_40_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_41__092_060open_062high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_060_Alength_AtreeList_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),size_s6755466524823107622T_VEBT(treeList)) ).

% \<open>high (summin * 2 ^ n + lx) n < length treeList\<close>
tff(fact_42_hlbound,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)) ) ).

% hlbound
tff(fact_43_nothprolist,axiom,
    ! [I2: nat] :
      ( ( ( I2 != 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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)) )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I2) = nth_VEBT_VEBT(treeList,I2) ) ) ).

% nothprolist
tff(fact_44_newlistlength,axiom,
    size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% newlistlength
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( member_real(A,collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_46_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(A,collec213857154873943460nt_int(P))
    <=> aa_Pro4592068669816069595_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_47_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( member_complex(A,collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_48_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_49_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_50_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( member_int(A,collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_51_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_52_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_53_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_54_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_55_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_56_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_57_Collect__cong,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X2)
        <=> aa_Pro4592068669816069595_int_o(Q,X2) )
     => ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).

% Collect_cong
tff(fact_58_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X2: complex] :
          ( aa_complex_o(P,X2)
        <=> aa_complex_o(Q,X2) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_59_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
        <=> aa_set_nat_o(Q,X2) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_60_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_61_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(Q,X2) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_62__C114_C,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,
          $ite(
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
            $let(
              maxs: option_nat,
              maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
              $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
            ma )),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),
        $ite(
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
          $let(
            maxs: option_nat,
            maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
            $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
          ma )) ) ).

% "114"
tff(fact_63__C112_C,axiom,
    ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = $ite(
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
          $let(
            maxs: option_nat,
            maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
            $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
          ma ) )
   => ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))))
       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) ) ).

% "112"
tff(fact_64__092_060open_062_092_060lbrakk_062_092_060forall_062t_092_060in_062set_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_O_Ainvar__vebt_At_An_059_Ainvar__vebt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Am_059_Alength_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_A_061_A2_A_094_Am_059_Am_A_061_ASuc_An_059_Adeg_A_061_An_A_L_Am_059_A_092_060forall_062i_0602_A_094_Am_O_A_I_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Ai_J_Ax_J_A_061_Aboth__member__options_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ai_059_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_A_092_060longrightarrow_062_A_I_092_060forall_062t_092_060in_062set_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_O_A_092_060nexists_062x_O_Aboth__member__options_At_Ax_J_059_Asummin_A_K_A2_A_094_An_A_L_Alx_A_092_060le_062_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_059_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_A_060_A2_A_094_Adeg_059_Asummin_A_K_A2_A_094_An_A_L_Alx_A_092_060noteq_062_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_A_092_060longrightarrow_062_A_I_092_060forall_062i_0602_A_094_Am_O_A_Ihigh_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_An_A_061_Ai_A_092_060longrightarrow_062_Aboth__member__options_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Ai_J_A_Ilow_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_An_J_J_A_092_060and_062_A_I_092_060forall_062x_O_Ahigh_Ax_An_A_061_Ai_A_092_060and_062_Aboth__member__options_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Ai_J_A_Ilow_Ax_An_J_A_092_060longrightarrow_062_Asummin_A_K_A2_A_094_An_A_L_Alx_A_060_Ax_A_092_060and_062_Ax_A_092_060le_062_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ainvar__vebt_A_INode_A_ISome_A_Isummin_A_K_A2_A_094_An_A_L_Alx_M_Aif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_Adeg_092_060close_062,axiom,
    ( ! [X2: vEBT_VEBT] :
        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))))
       => aa_nat_o(vEBT_invar_vebt(X2),na) )
   => ( aa_nat_o(vEBT_invar_vebt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),m)
     => ( ( size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))) = 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,na) )
         => ( ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
           => ( ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
                 => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I3)),X_12)
                  <=> aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I3) ) )
             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = $ite(
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                        $let(
                          maxs: option_nat,
                          maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                          $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                        ma ) )
                 => ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),
                      $ite(
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                        $let(
                          maxs: option_nat,
                          maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                          $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                        ma ))
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,
                          $ite(
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                            $let(
                              maxs: option_nat,
                              maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                              $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                            ma )),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))
                   => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) != $ite(
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                              $let(
                                maxs: option_nat,
                                maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                                $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                              ma ) )
                       => ! [I3: nat] :
                            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
                           => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,
                                        $ite(
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                                          $let(
                                            maxs: option_nat,
                                            maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                                            $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                                          ma )),na) = I3 )
                               => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I3)),
                                    aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,
                                        $ite(
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                                          $let(
                                            maxs: option_nat,
                                            maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                                            $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                                          ma )),na)) )
                              & ! [X2: nat] :
                                  ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),na) = I3 )
                                    & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),na)) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),X2)
                                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),
                                        $ite(
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                                          $let(
                                            maxs: option_nat,
                                            maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                                            $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                                          ma )) ) ) ) ) )
                     => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),
                                  $ite(
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                                      $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                                    ma ))),deg,list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))),deg) ) ) ) ) ) ) ) ) ) ) ).

% \<open>\<lbrakk>\<forall>t\<in>set (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)]). invar_vebt t n; invar_vebt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) m; length (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)]) = 2 ^ m; m = Suc n; deg = n + m; \<forall>i<2 ^ m. (\<exists>x. both_member_options (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! i) x) = both_member_options (vebt_delete summary (high (summin * 2 ^ n + lx) n)) i; summin * 2 ^ n + lx = (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma) \<longrightarrow> (\<forall>t\<in>set (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)]). \<nexists>x. both_member_options t x); summin * 2 ^ n + lx \<le> (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma); (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma) < 2 ^ deg; summin * 2 ^ n + lx \<noteq> (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma) \<longrightarrow> (\<forall>i<2 ^ m. (high (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma) n = i \<longrightarrow> both_member_options (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! i) (low (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma) n)) \<and> (\<forall>x. high x n = i \<and> both_member_options (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! i) (low x n) \<longrightarrow> summin * 2 ^ n + lx < x \<and> x \<le> (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma)))\<rbrakk> \<Longrightarrow> invar_vebt (Node (Some (summin * 2 ^ n + lx, if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma)) deg (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)]) (vebt_delete summary (high (summin * 2 ^ n + lx) n))) deg\<close>
tff(fact_65__C111_C,axiom,
    ! [I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I)),X_12)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I) ) ) ).

% "111"
tff(fact_66__C115_C,axiom,
    ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) != $ite(
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
          $let(
            maxs: option_nat,
            maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
            $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
          ma ) )
   => ! [I: nat] :
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,
                    $ite(
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                      $let(
                        maxs: option_nat,
                        maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                        $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                      ma )),na) = I )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I)),
                aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,
                    $ite(
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                      $let(
                        maxs: option_nat,
                        maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                        $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                      ma )),na)) )
          & ! [Y2: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y2),na) = I )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y2),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),Y2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),
                    $ite(
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                      $let(
                        maxs: option_nat,
                        maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                        $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                      ma )) ) ) ) ) ) ).

% "115"
tff(fact_67_add__self__div__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).

% add_self_div_2
tff(fact_68__C5_Ohyps_C_I7_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5.hyps"(7)
tff(fact_69_xnin,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)) ).

% xnin
tff(fact_70__C10_C,axiom,
    aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),deg) ).

% "10"
tff(fact_71_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_72_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_73_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_74_dsimp,axiom,
    vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),
          $ite(
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
            $let(
              maxs: option_nat,
              maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
              $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
            ma ))),deg,list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))) ).

% dsimp
tff(fact_75_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_76_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_77_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_78_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_79_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_80_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_81_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_82_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_83_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_84_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_85_power2__sum,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_86_power2__sum,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_87_power2__sum,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xb),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(Xb),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))),Xb)),Y)) ).

% power2_sum
tff(fact_88_power2__sum,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_num_nat(numeral_numeral_nat,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,Xb),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))),Xb)),Y)) ).

% power2_sum
tff(fact_89_power2__sum,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),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(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_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))),Xb)),Y)) ).

% power2_sum
tff(fact_90_False,axiom,
    xa = mi ).

% False
tff(fact_91_power__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_92_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb) ) ).

% maxbmo
tff(fact_93_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_1) ) ).

% not_min_Null_member
tff(fact_94_min__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),X3) ) ) ) ).

% min_in_set_def
tff(fact_95_max__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xb) ) ) ) ).

% max_in_set_def
tff(fact_96__C5_C,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) ) ).

% "5"
tff(fact_97_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,set_complex2(Xs))
         => aa_complex_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_98_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X2: real] :
          ( member_real(X2,set_real2(Xs))
         => aa_real_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_99_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Nb: nat] :
      ( ! [X2: set_nat] :
          ( member_set_nat(X2,set_set_nat2(Xs))
         => aa_set_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,nth_set_nat(Xs,Nb)) ) ) ).

% inthall
tff(fact_100_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_101_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_102_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_o_o(P,(X2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_103_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X2: int] :
          ( member_int(X2,set_int2(Xs))
         => aa_int_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_104_xmi,axiom,
    xa = mi ).

% xmi
tff(fact_105_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Mb) = numera6690914467698888265omplex(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_106_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Mb) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_107_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Mb) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_108_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Mb) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_109_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Mb) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_110_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X22) = some_P7363390416028606310at_nat(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_111_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( some_nat(X22) = some_nat(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_112_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( some_num(X22) = some_num(Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_113__092_060open_062x_A_092_060noteq_062_Ami_A_092_060or_062_Ax_A_092_060noteq_062_Ama_092_060close_062,axiom,
    ( ( xa != mi )
    | ( xa != ma ) ) ).

% \<open>x \<noteq> mi \<or> x \<noteq> ma\<close>
tff(fact_114_inrg,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),xa)
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,xa),ma) ) ).

% inrg
tff(fact_115_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_116_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_117_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_118_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_119_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_120_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_121_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_122_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_123_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_124_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_125_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_126_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_127_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_128_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_129_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_130_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_131_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_132_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% numeral_times_numeral
tff(fact_133_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_134_add__numeral__left,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_135_add__numeral__left,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_136_add__numeral__left,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_137_add__numeral__left,axiom,
    ! [V: num,W: 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,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_138_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_139_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_140_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_141_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_142_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_143_num__double,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_144_del__single__cont,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ) ) ) ).

% del_single_cont
tff(fact_145_power__mult__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_146_power__mult__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_147_power__mult__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_148_power__mult__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% power_mult_numeral
tff(fact_149_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y3: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y3)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_150_not__Some__eq,axiom,
    ! [Xb: option_nat] :
      ( ! [Y3: nat] : Xb != some_nat(Y3)
    <=> ( Xb = none_nat ) ) ).

% not_Some_eq
tff(fact_151_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y3: num] : Xb != some_num(Y3)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_152_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y3: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y3) ) ).

% not_None_eq
tff(fact_153_not__None__eq,axiom,
    ! [Xb: option_nat] :
      ( ( Xb != none_nat )
    <=> ? [Y3: nat] : Xb = some_nat(Y3) ) ).

% not_None_eq
tff(fact_154_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y3: num] : Xb = some_num(Y3) ) ).

% not_None_eq
tff(fact_155__C7_C,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [Y2: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y2),na) = I )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y2),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),Y2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),ma) ) ) ) ) ) ).

% "7"
tff(fact_156_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(Xb)),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_157__C6_C,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma)
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ) ).

% "6"
tff(fact_158_power__add__numeral2,axiom,
    ! [A: complex,Mb: num,Nb: num,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_159_power__add__numeral2,axiom,
    ! [A: real,Mb: num,Nb: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_160_power__add__numeral2,axiom,
    ! [A: rat,Mb: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_161_power__add__numeral2,axiom,
    ! [A: nat,Mb: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_162_power__add__numeral2,axiom,
    ! [A: int,Mb: num,Nb: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_163_power__add__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_164_power__add__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_165_power__add__numeral,axiom,
    ! [A: rat,Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_166_power__add__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_167_power__add__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% power_add_numeral
tff(fact_168_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% Suc_numeral
tff(fact_169__092_060open_062mi_A_092_060noteq_062_Ama_A_092_060and_062_Ax_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    ( ( mi != ma )
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ) ).

% \<open>mi \<noteq> ma \<and> x < 2 ^ deg\<close>
tff(fact_170_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_171_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_172_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_173_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_174_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_175_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_176_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_177_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_178__092_060open_062Some_Asummin_A_061_Avebt__mint_Asummary_092_060close_062,axiom,
    some_nat(summin) = vEBT_vebt_mint(summary) ).

% \<open>Some summin = vebt_mint summary\<close>
tff(fact_179_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_180_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_181_yhelper,axiom,
    ! [Y: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y),na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),Y)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),ma)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)) ) ) ) ).

% yhelper
tff(fact_182__C7b_C,axiom,
    ! [I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I )
         => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
        & ! [Y2: nat] :
            ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Y2),na) = I )
              & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Y2),na)) )
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),Y2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),ma) ) ) ) ) ).

% "7b"
tff(fact_183_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_184_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                $ite(Xb = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_185_del__x__not__mia,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: nat,H: nat,L: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
               => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(
                                    Xb = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                $ite(Xb = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_186_del__x__not__mi__new__node__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Sn: vEBT_VEBT,Summarya: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(
                                    Xb = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_187_del__x__not__mi,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                   => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,H),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(
                                    Xb = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,Newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                $ite(Xb = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_188__092_060open_062Some_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_092_060close_062,axiom,
    some_nat(lx) = vEBT_vebt_mint(nth_VEBT_VEBT(treeList,summin)) ).

% \<open>Some lx = vebt_mint (treeList ! summin)\<close>
tff(fact_189_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one) ).

% add_One_commute
tff(fact_190_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_191_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_192_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_193_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_194_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_195_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_196_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_197_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_198_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_199_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_200_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),A) ).

% power_Suc2
tff(fact_201_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).

% power_Suc2
tff(fact_202_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_203_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_204_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),A) ).

% power_Suc2
tff(fact_205_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_Suc
tff(fact_206_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_Suc
tff(fact_207_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_208_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_209_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_Suc
tff(fact_210_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),I2) ) ).

% less_mult_imp_div_less
tff(fact_211_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_212_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_213_power__commuting__commutes,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Xb) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_214_power__commuting__commutes,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_215_power__commuting__commutes,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y) = aa_rat_rat(times_times_rat(Y),Xb) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_216_power__commuting__commutes,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xb) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_217_power__commuting__commutes,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_218_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),Nb)) ).

% power_mult_distrib
tff(fact_219_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_220_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_221_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_222_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ).

% power_mult_distrib
tff(fact_223_power__commutes,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_commutes
tff(fact_224_power__commutes,axiom,
    ! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_commutes
tff(fact_225_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_226_power__commutes,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_commutes
tff(fact_227_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_commutes
tff(fact_228_power__divide,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),Nb)) ).

% power_divide
tff(fact_229_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_230_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),B)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_231_power__mult,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),Nb) ).

% power_mult
tff(fact_232_power__mult,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),Nb) ).

% power_mult
tff(fact_233_power__mult,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),Nb) ).

% power_mult
tff(fact_234_power__mult,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Mb)),Nb) ).

% power_mult
tff(fact_235_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_236_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B2: nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_237_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
       => ( ! [A3: product_prod_nat_nat,B2: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A3) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_238_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
       => ( ! [A3: nat,B2: product_prod_nat_nat] :
              ( ( Xb = some_nat(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_239_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( Xb = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
       => ( ! [A3: nat,B2: nat] :
              ( ( Xb = some_nat(A3) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_240_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( Xb = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
       => ( ! [A3: nat,B2: num] :
              ( ( Xb = some_nat(A3) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_241_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
       => ( ! [A3: num,B2: product_prod_nat_nat] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_242_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( Xb = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
       => ( ! [A3: num,B2: nat] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_243_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
       => ( ! [A3: num,B2: num] :
              ( ( Xb = some_num(A3) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_244_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_all
tff(fact_245_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X3: nat] : aa_option_nat_o(P,some_nat(X3)) ) ) ).

% split_option_all
tff(fact_246_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_all
tff(fact_247_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_ex
tff(fact_248_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X3: nat] : aa_option_nat_o(P,some_nat(X3)) ) ) ).

% split_option_ex
tff(fact_249_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_ex
tff(fact_250_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_251_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_252_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_253_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_254_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = some_nat(X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_255_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = some_num(X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_256_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_257_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != some_nat(X22) ).

% option.distinct(1)
tff(fact_258_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != some_num(X22) ).

% option.distinct(1)
tff(fact_259_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q2) ).

% div_mult2_eq
tff(fact_260_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_261_option_Osel,axiom,
    ! [X22: nat] : the_nat(some_nat(X22)) = X22 ).

% option.sel
tff(fact_262_option_Osel,axiom,
    ! [X22: num] : the_num(some_num(X22)) = X22 ).

% option.sel
tff(fact_263_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_264_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_265_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_266_less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% less_exp
tff(fact_267_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_268_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_269_self__le__ge2__pow,axiom,
    ! [K: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_270_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_271_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_272_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_273_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_274_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_275_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_276_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_277_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_278_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_279_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_280_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_281_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_282_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_283_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_284_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_285_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_286_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_287_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_288_power__add,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_add
tff(fact_289_power__add,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_290_power__add,axiom,
    ! [A: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_291_power__add,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_292_power__add,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_add
tff(fact_293_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_294_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_295_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_296_power__odd__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_297_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_298_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_299_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_300_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_301_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,m2),m2) ) ).

% numeral_code(2)
tff(fact_302_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2) ) ).

% numeral_code(2)
tff(fact_303_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(m2),m2) ) ).

% numeral_code(2)
tff(fact_304_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_305_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_306_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_307_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_308_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,w),w) ) ).

% power_numeral_even
tff(fact_309_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_310_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(w),w) ) ).

% power_numeral_even
tff(fact_311_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_312_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_313_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_314_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_315_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_316_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_317_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_318_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),Z) ).

% mult_2_right
tff(fact_319_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_320_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_321_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_322_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_323_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),Z) ).

% mult_2
tff(fact_324_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_325_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_326_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_327_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_328_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),A) ).

% power2_eq_square
tff(fact_329_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).

% power2_eq_square
tff(fact_330_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A),A) ).

% power2_eq_square
tff(fact_331_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_332_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_333_power4__eq__xxxx,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_334_power4__eq__xxxx,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = 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,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_335_power4__eq__xxxx,axiom,
    ! [Xb: rat] : aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_336_power4__eq__xxxx,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_337_power4__eq__xxxx,axiom,
    ! [Xb: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_338_power__even__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_339_power__even__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_340_power__even__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_341_power__even__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_342_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_343_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Nb) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_344_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),Nb) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_345_succ__list__to__short,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_346_pred__list__to__short,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_347_del__x__mia,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                xn: nat,
                xn:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,h),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                  $ite(
                                    xn = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Maa ))),Dega,newlist,sn) ),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Dega,newlist,Summarya) ) ) ),
                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_348_del__x__mi__lets__in__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_349_del__x__mi__lets__in,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,Newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_350_del__x__mi,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,H),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(
                                      Xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Maa))),Dega,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_351_del__in__range,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                xn: nat,
                xn:= 
                  $ite(Xb = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xb),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xb = Mia,xn,Mia),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,h),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(
                                      xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Maa ))),Dega,newlist,sn) ),
                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                  $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),Dega,newlist,Summarya) ) ) ),
                      vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_352__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_061_A_Iif_Ax_A_061_Ami_Athen_Athe_A_Ivebt__mint_Asummary_J_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_Aelse_Ax_J_092_060close_062,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = $ite(xa = mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_mint(summary)))))),xa) ).

% \<open>summin * 2 ^ n + lx = (if x = mi then the (vebt_mint summary) * 2 ^ (deg div 2) + the (vebt_mint (treeList ! the (vebt_mint summary))) else x)\<close>
tff(fact_353_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                  $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Maa))),Dega,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_354_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_355_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_356__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062summin_O_ASome_Asummin_A_061_Avebt__mint_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Summin: nat] : some_nat(Summin) != vEBT_vebt_mint(summary) ).

% \<open>\<And>thesis. (\<And>summin. Some summin = vebt_mint summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_357_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_358__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lx_O_ASome_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Lx: nat] : some_nat(Lx) != vEBT_vebt_mint(nth_VEBT_VEBT(treeList,summin)) ).

% \<open>\<And>thesis. (\<And>lx. Some lx = vebt_mint (treeList ! summin) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_359_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Maa),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_360_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( some_nat(Mb) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% misiz
tff(fact_361_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpyd
tff(fact_362_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = some_nat(Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpypredd
tff(fact_363_pred__max,axiom,
    ! [Dega: nat,Maa: nat,Xb: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Maa) ) ) ) ).

% pred_max
tff(fact_364_succ__min,axiom,
    ! [Dega: nat,Xb: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Mia) ) ) ) ).

% succ_min
tff(fact_365_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Va: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Mia)
         => ( ( Maa != Mia )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_366_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_367_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),one)
    <=> ( Xb = one ) ) ).

% le_num_One_iff
tff(fact_368_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_369_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_370_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% mint_corr
tff(fact_371_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_mint(Ta) = some_nat(Xb) ) ) ) ).

% mint_sound
tff(fact_372_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% maxt_corr
tff(fact_373_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_maxt(Ta) = some_nat(Xb) ) ) ) ).

% maxt_sound
tff(fact_374_vebt__delete_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb) ),
        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),
        $ite(
          ( ( Xb = Mia )
          & ( Xb = Maa ) ),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(vEBT_vebt_mint(Summarya)))))),Xb),
            $let(
              minn: nat,
              minn:= 
                $ite(Xb = Mia,xn,Mia),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summarya,h),
                          vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                $ite(
                                  xn = Maa,
                                  $let(
                                    maxs: option_nat,
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                  Maa ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,sn) ),
                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                              $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist,Summarya) ) ) ),
                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_375_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_376_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_377_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_378_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_379_set__swap,axiom,
    ! [I2: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I2,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I2))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_380_set__swap,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I2))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_381_set__swap,axiom,
    ! [I2: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,I2,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I2))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_382_set__swap,axiom,
    ! [I2: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,I2,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I2))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_383_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_384_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_385_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_386_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_387_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_388__C11_C,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),na) ).

% "11"
tff(fact_389_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_390_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_391_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,Xb: vEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,Xb),I2,Y) = list_u1324408373059187874T_VEBT(Xs,I2,Y) ).

% list_update_overwrite
tff(fact_392_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_393_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_394__092_060open_062vebt__member_A_ItreeList_A_B_Asummin_J_Alx_092_060close_062,axiom,
    aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,summin)),lx) ).

% \<open>vebt_member (treeList ! summin) lx\<close>
tff(fact_395_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_396__092_060open_062vebt__member_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
    aa_nat_o(vEBT_vebt_member(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na)) ).

% \<open>vebt_member summary (high ma n)\<close>
tff(fact_397_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_398_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_399_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_400_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_401_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_402_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_403_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_404_power__one__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),one_one_nat) = A ).

% power_one_right
tff(fact_405_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),one_one_nat) = A ).

% power_one_right
tff(fact_406_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_407_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,Xb)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_408_length__list__update,axiom,
    ! [Xs: list_o,I2: nat,Xb: $o] : size_size_list_o(list_update_o(Xs,I2,(Xb))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_409_length__list__update,axiom,
    ! [Xs: list_int,I2: nat,Xb: int] : size_size_list_int(list_update_int(Xs,I2,Xb)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_410_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_411_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_nat,Xb: nat] :
      ( ( I2 != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_412_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_int,Xb: int] :
      ( ( I2 != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I2,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_413_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( I2 != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_414_list__update__id,axiom,
    ! [Xs: list_nat,I2: nat] : list_update_nat(Xs,I2,aa_nat_nat(nth_nat(Xs),I2)) = Xs ).

% list_update_id
tff(fact_415_list__update__id,axiom,
    ! [Xs: list_int,I2: nat] : list_update_int(Xs,I2,aa_nat_int(nth_int(Xs),I2)) = Xs ).

% list_update_id
tff(fact_416_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat] : list_u1324408373059187874T_VEBT(Xs,I2,nth_VEBT_VEBT(Xs,I2)) = Xs ).

% list_update_id
tff(fact_417_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_418_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X2),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_419_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_420_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_421_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_422_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_423_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_424_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_425_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_426_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_427_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_428_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_429_power__inject__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_430_power__inject__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Mb) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_431_power__inject__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_432_power__inject__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_433_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I2: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I2)
     => ( list_u1324408373059187874T_VEBT(Xs,I2,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_434_list__update__beyond,axiom,
    ! [Xs: list_o,I2: nat,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I2)
     => ( list_update_o(Xs,I2,(Xb)) = Xs ) ) ).

% list_update_beyond
tff(fact_435_list__update__beyond,axiom,
    ! [Xs: list_int,I2: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I2)
     => ( list_update_int(Xs,I2,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_436_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_437_power__strict__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_438_power__strict__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Xb)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_439_power__strict__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_440_power__strict__increasing__iff,axiom,
    ! [B: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_441_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,Xb)),I2) = Xb ) ) ).

% nth_list_update_eq
tff(fact_442_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,Xb),I2) = Xb ) ) ).

% nth_list_update_eq
tff(fact_443_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I2,(Xb))),I2)
      <=> (Xb) ) ) ).

% nth_list_update_eq
tff(fact_444_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I2,Xb)),I2) = Xb ) ) ).

% nth_list_update_eq
tff(fact_445_one__add__one,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_446_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_447_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_448_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_449_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_450_power__increasing__iff,axiom,
    ! [B: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_451_power__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Xb)),aa_nat_rat(power_power_rat(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_452_power__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_453_power__increasing__iff,axiom,
    ! [B: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_454_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_455_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_456_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_457_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_458_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_459_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% one_plus_numeral
tff(fact_460_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_461_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_462_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_463_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_464_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% numeral_plus_one
tff(fact_465_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_466_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_467_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_468_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_469_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_470_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_471_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_472_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_473_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [Z2: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),Xb) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),Y) ) ) ) ).

% pred_member
tff(fact_474_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [Z2: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z2) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z2) ) ) ) ).

% succ_member
tff(fact_475_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Sx) ) ) ).

% succ_corr
tff(fact_476_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Px: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Px) ) ) ).

% pred_corr
tff(fact_477_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_478_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_479_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_480_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_481_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_482_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_483_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_484_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_485_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_486_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_487_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_488_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_489_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_490_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_491_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_492_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_493_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_494_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_495_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_496_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_497_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_498_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),numera6690914467698888265omplex(Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Xb)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_499_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_500_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_501_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_502_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_503_one__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% one_le_power
tff(fact_504_one__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% one_le_power
tff(fact_505_one__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% one_le_power
tff(fact_506_one__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% one_le_power
tff(fact_507_left__right__inverse__power,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y) = one_one_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_508_left__right__inverse__power,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_509_left__right__inverse__power,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_510_left__right__inverse__power,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_511_left__right__inverse__power,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_512_power__one__over,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),A)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_one_over
tff(fact_513_power__one__over,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_one_over
tff(fact_514_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),Nb) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_515_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_516_power__less__power__Suc,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_517_power__less__power__Suc,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_518_power__less__power__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_519_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_520_power__gt1__lemma,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_521_power__gt1__lemma,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_522_power__gt1__lemma,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_523_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_524_power__gt1,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_525_power__gt1,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_526_power__gt1,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_527_power__gt1,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_528_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_529_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_530_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_531_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_532_power__less__imp__less__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_533_power__less__imp__less__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_534_power__less__imp__less__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_535_power__less__imp__less__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_536_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)) ) ) ).

% power_increasing
tff(fact_537_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_increasing
tff(fact_538_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_increasing
tff(fact_539_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ).

% power_increasing
tff(fact_540_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B3)
    <=> ! [X3: complex] :
          ( member_complex(X3,set_complex2(Xs))
         => member_complex(X3,B3) ) ) ).

% subset_code(1)
tff(fact_541_subset__code_I1_J,axiom,
    ! [Xs: list_real,B3: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B3)
    <=> ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => member_real(X3,B3) ) ) ).

% subset_code(1)
tff(fact_542_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),B3)
    <=> ! [X3: set_nat] :
          ( member_set_nat(X3,set_set_nat2(Xs))
         => member_set_nat(X3,B3) ) ) ).

% subset_code(1)
tff(fact_543_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B3)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X3),B3) ) ) ).

% subset_code(1)
tff(fact_544_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B3)
    <=> ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X3,B3) ) ) ).

% subset_code(1)
tff(fact_545_subset__code_I1_J,axiom,
    ! [Xs: list_int,B3: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B3)
    <=> ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => member_int(X3,B3) ) ) ).

% subset_code(1)
tff(fact_546_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_547_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_548_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_549_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_550_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_551_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_552_list__update__swap,axiom,
    ! [I2: nat,I4: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,X4: vEBT_VEBT] :
      ( ( I2 != I4 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,Xb),I4,X4) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I4,X4),I2,Xb) ) ) ).

% list_update_swap
tff(fact_553_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_554_power__le__imp__le__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_555_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_556_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_557_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_558_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_559_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_560_one__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_561_one__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_562_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_563_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_564_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_565_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_566_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,Xb: complex,I2: nat] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),A2)
     => ( member_complex(Xb,A2)
       => ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I2,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_567_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,Xb: real,I2: nat] :
      ( ord_less_eq_set_real(set_real2(Xs),A2)
     => ( member_real(Xb,A2)
       => ord_less_eq_set_real(set_real2(list_update_real(Xs,I2,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_568_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,Xb: set_nat,I2: nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),A2)
     => ( member_set_nat(Xb,A2)
       => ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I2,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_569_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,Xb: nat,I2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A2)
     => ( aa_set_nat_o(member_nat(Xb),A2)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I2,Xb))),A2) ) ) ).

% set_update_subsetI
tff(fact_570_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,Xb: vEBT_VEBT,I2: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A2)
     => ( member_VEBT_VEBT(Xb,A2)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I2,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_571_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,Xb: int,I2: nat] :
      ( ord_less_eq_set_int(set_int2(Xs),A2)
     => ( member_int(Xb,A2)
       => ord_less_eq_set_int(set_int2(list_update_int(Xs,I2,Xb)),A2) ) ) ).

% set_update_subsetI
tff(fact_572_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_573_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_574_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] :
            ( aa_nat_o(aa_nat_fun_nat_o(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_575_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] :
            ( aa_nat_o(aa_nat_fun_nat_o(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_576_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I5)
            <=> aa_nat_o(nth_o(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_577_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] :
            ( aa_nat_o(aa_nat_fun_nat_o(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_578_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I5),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I5),aa_nat_nat(nth_nat(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_579_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I5),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I5),nth_VEBT_VEBT(Xs3,I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_580_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I5),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_o_o(aa_nat_fun_o_o(P,I5),aa_nat_o(nth_o(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_581_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I5),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_int_o(aa_nat_fun_int_o(P,I5),aa_nat_int(nth_int(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_582_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_583_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys,I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_584_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_585_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_586_vebt__delete_Osimps_I4_J,axiom,
    ! [Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya),Uu) = vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya) ).

% vebt_delete.simps(4)
tff(fact_587_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_588_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_589_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X3: $o] :
          ( member_o((X3),set_o2(Xs))
         => aa_o_o(P,(X3)) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_590_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => aa_int_o(P,X3) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_591_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xb: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I3)) )
     => ( member_complex(Xb,set_complex2(Xs))
       => aa_complex_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_592_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xb: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I3)) )
     => ( member_real(Xb,set_real2(Xs))
       => aa_real_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_593_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Xb: set_nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,nth_set_nat(Xs,I3)) )
     => ( member_set_nat(Xb,set_set_nat2(Xs))
       => aa_set_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_594_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xb: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) )
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
       => aa_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_595_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) )
     => ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_596_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xb: $o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) )
     => ( member_o((Xb),set_o2(Xs))
       => aa_o_o(P,(Xb)) ) ) ).

% all_nth_imp_all_set
tff(fact_597_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) )
     => ( member_int(Xb,set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_598_in__set__conv__nth,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,set_complex2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_599_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_600_in__set__conv__nth,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xb,set_set_nat2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s3254054031482475050et_nat(Xs))
          & ( nth_set_nat(Xs,I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_601_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_602_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_603_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I5)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_604_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( member_int(Xb,set_int2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_605_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
           => aa_nat_o(P,X2) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_606_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X2: vEBT_VEBT] :
            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X2) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_607_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X2: $o] :
            ( member_o((X2),set_o2(Xs))
           => aa_o_o(P,(X2)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_608_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X2: int] :
            ( member_int(X2,set_int2(Xs))
           => aa_int_o(P,X2) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_609_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(aa_nat_complex(nth_complex(Xs),Nb),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_610_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Nb),set_real2(Xs)) ) ).

% nth_mem
tff(fact_611_nth__mem,axiom,
    ! [Nb: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(nth_set_nat(Xs,Nb),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_612_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_613_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_614_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Nb),set_o2(Xs)) ) ).

% nth_mem
tff(fact_615_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(aa_nat_int(nth_int(Xs),Nb),set_int2(Xs)) ) ).

% nth_mem
tff(fact_616_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(Xb,set_complex2(list_update_complex(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_617_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(Xb,set_real2(list_update_real(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_618_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_set_nat,Xb: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(Xb,set_set_nat2(list_update_set_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_619_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_620_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(Xb,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_621_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o((Xb),set_o2(list_update_o(Xs,Nb,(Xb)))) ) ).

% set_update_memI
tff(fact_622_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(Xb,set_int2(list_update_int(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_623_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I2,Xb) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I2) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_624_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I2,Xb) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I2) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_625_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I2,(Xb)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I2)
        <=> (Xb) ) ) ) ).

% list_update_same_conv
tff(fact_626_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I2,Xb) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I2) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_627_nth__list__update,axiom,
    ! [I2: nat,Xs: list_nat,Xb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I2,Xb)),J) = $ite(I2 = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_628_nth__list__update,axiom,
    ! [I2: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I2,Xb),J) = $ite(I2 = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_629_nth__list__update,axiom,
    ! [I2: nat,Xs: list_o,Xb: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I2,(Xb))),J)
      <=> $ite(I2 = J,(Xb),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_630_nth__list__update,axiom,
    ! [I2: nat,Xs: list_int,Xb: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I2,Xb)),J) = $ite(I2 = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_631_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_632_greater__shift,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(Xb)),some_nat(Y)) ) ).

% greater_shift
tff(fact_633_less__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(Xb)),some_nat(Y)) ) ).

% less_shift
tff(fact_634_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_635_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_636_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
          & ? [M: nat] :
              ( ( some_nat(M) = vEBT_vebt_mint(Summarya) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_637_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).

% semiring_norm(69)
tff(fact_638_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_639_sum__squares__bound,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_640_sum__squares__bound,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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_641_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_642_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_643_semiring__norm_I87_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit0,Mb) = aa_num_num(bit0,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(87)
tff(fact_644_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_645_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_646_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(Ta)),Xb),Sx) ) ) ).

% pred_correct
tff(fact_647_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(Ta)),Xb),Sx) ) ) ).

% succ_correct
tff(fact_648_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).

% power_minus_is_div
tff(fact_649_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa_num_num(bit0,Mb) != one ).

% semiring_norm(85)
tff(fact_650_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_651_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_652_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_653_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_654_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_655_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_656_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_657_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_658_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_659_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_660_diff__diff__left,axiom,
    ! [I2: 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,I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_661_diff__diff__cancel,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_662_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_663_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_664_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(6)
tff(fact_665_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_666_semiring__norm_I13_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% semiring_norm(13)
tff(fact_667_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_668_semiring__norm_I11_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Mb),one) = Mb ).

% semiring_norm(11)
tff(fact_669_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_670_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_671_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),one) ).

% semiring_norm(75)
tff(fact_672_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_673_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_674_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_675_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_676_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_677_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_678_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_679_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_680_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_681_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = $ite(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_682_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(1)
tff(fact_683_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(1)
tff(fact_684_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_685_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_686_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_687_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(5)
tff(fact_688_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Xb)) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(5)
tff(fact_689_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(5)
tff(fact_690_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(5)
tff(fact_691_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(5)
tff(fact_692_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(5)
tff(fact_693_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(6)
tff(fact_694_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Xb),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(6)
tff(fact_695_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(6)
tff(fact_696_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(6)
tff(fact_697_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(6)
tff(fact_698_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(6)
tff(fact_699_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),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,I2),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_700_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_701_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),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,I2),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_702_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_703_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),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,I2),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_704_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_705_diff__commute,axiom,
    ! [I2: 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,I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),K)),J) ).

% diff_commute
tff(fact_706_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_707_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_708_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ).

% nat_add_max_right
tff(fact_709_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_710_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)) ).

% nat_mult_max_right
tff(fact_711_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I2: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I2)) ) ) ).

% zero_induct_lemma
tff(fact_712_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_713_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_714_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_715_diff__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_cancel2
tff(fact_716_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_717_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_718_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_719_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_720_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_721_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_722_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_723_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_724_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_725_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% diff_mult_distrib
tff(fact_726_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% diff_mult_distrib2
tff(fact_727_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_728_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_729_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_730_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),J) ) ).

% less_diff_conv
tff(fact_731_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_732_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_733_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_734_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)) ) ).

% le_diff_conv
tff(fact_735_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_736_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_737_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),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)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_738_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_739_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_740_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)) ) ) ).

% less_diff_conv2
tff(fact_741_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_742_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_743_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_744_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_745_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_746_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_747_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_748_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_749_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_750_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_751_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_752_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_753_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_754_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_755_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_756_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_757_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_758_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_759_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => ? [X2: nat] :
            ( aa_nat_o(P,X2)
            & ! [Y2: nat] :
                ( aa_nat_o(P,Y2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_760_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_761_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_762_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_763_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K) ) ) ).

% le_trans
tff(fact_764_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_765_size__neq__size__imp__neq,axiom,
    ! [Xb: char,Y: char] :
      ( ( size_size_char(Xb) != size_size_char(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_766_size__neq__size__imp__neq,axiom,
    ! [Xb: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xb) != size_s6755466524823107622T_VEBT(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_767_size__neq__size__imp__neq,axiom,
    ! [Xb: list_o,Y: list_o] :
      ( ( size_size_list_o(Xb) != size_size_list_o(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_768_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Y: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_769_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_770_power2__commute,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_771_power2__commute,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_772_power2__commute,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_773_power2__commute,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_774_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_775_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_776_power2__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_777_power2__diff,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_778_power2__diff,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),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(Xb),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))),Xb)),Y)) ).

% power2_diff
tff(fact_779_power2__diff,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),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(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_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))),Xb)),Y)) ).

% power2_diff
tff(fact_780_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_781_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => aa_nat_o(P,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I3))
               => aa_nat_o(P,I3) ) )
         => aa_nat_o(P,I2) ) ) ) ).

% strict_inc_induct
tff(fact_782_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I3),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I2),J) ) ) ) ).

% less_Suc_induct
tff(fact_783_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),K) ) ) ).

% less_trans_Suc
tff(fact_784_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_785_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_786_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M3: nat] :
          ( ( Mb = aa_nat_nat(suc,M3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M3) ) ) ).

% Suc_less_eq2
tff(fact_787_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
           => aa_nat_o(P,I5) ) ) ) ).

% All_less_Suc
tff(fact_788_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_789_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_790_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
            & aa_nat_o(P,I5) ) ) ) ).

% Ex_less_Suc
tff(fact_791_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_792_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_793_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_794_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_795_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_796_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
     => ( ( K != aa_nat_nat(suc,I2) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_797_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_798_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_799_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_800_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X2),X2)
       => ( ! [X2: nat,Y4: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y4)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y4),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X2),Z3) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_801_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P,Mb)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_802_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_803_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_804_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_805_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_806_Suc__le__D,axiom,
    ! [Nb: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M4)
     => ? [M: nat] : M4 = aa_nat_nat(suc,M) ) ).

% Suc_le_D
tff(fact_807_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_808_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_809_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_810_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_811_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K) ) ).

% add_lessD1
tff(fact_812_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_813_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),I2) ).

% not_add_less1
tff(fact_814_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),I2) ).

% not_add_less2
tff(fact_815_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_816_trans__less__add1,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_less_add1
tff(fact_817_trans__less__add2,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_less_add2
tff(fact_818_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_819_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_820_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_821_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_822_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_823_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_824_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_825_add__leE,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_826_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_827_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_828_add__leD1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_829_add__leD2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_830_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_831_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_832_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_833_trans__le__add1,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_le_add1
tff(fact_834_trans__le__add2,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_le_add2
tff(fact_835_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3) ) ).

% nat_le_iff_add
tff(fact_836_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% add_mult_distrib
tff(fact_837_add__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% add_mult_distrib2
tff(fact_838_left__add__mult__distrib,axiom,
    ! [I2: 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,I2),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),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,I2),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_839_le__cube,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).

% le_cube
tff(fact_840_le__square,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).

% le_square
tff(fact_841_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_842_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_843_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_844_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_845_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_846_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(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,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_847_four__x__squared,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_848_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_849_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_850_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_851_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_852_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_853_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_854_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_855_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_856_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_857_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_858_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N)),aa_nat_set_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ord_less_eq_set_int(aa_nat_set_int(F,N3),aa_nat_set_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_859_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_860_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_861_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_862_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_863_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(F,N),aa_nat_set_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ord_less_eq_set_int(aa_nat_set_int(F,Nb),aa_nat_set_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_864_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_865_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_866_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_867_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_868_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ) ).

% less_natE
tff(fact_869_less__add__Suc1,axiom,
    ! [I2: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Mb))) ).

% less_add_Suc1
tff(fact_870_less__add__Suc2,axiom,
    ! [I2: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),I2))) ).

% less_add_Suc2
tff(fact_871_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_872_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)) ) ).

% less_imp_Suc_add
tff(fact_873_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_874_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_875_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_876_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_877_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_878_inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I2) ) ) ) ).

% inc_induct
tff(fact_879_dec__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(P,I2)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_880_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_881_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_882_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_883_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_884_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_885_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K: nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Mb)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_886_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_887_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_888_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_889_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_890_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_891_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_892_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_893_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_894_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_895_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_896_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_897_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_898_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_899_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_900_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_901_pred__less__length__list,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_902_pred__lesseq__max,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maa)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_903_succ__greatereq__min,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_904_succ__less__length__list,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_905_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_906_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_907_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_908_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_909_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_910_add__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_911_mul__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_912_real__divide__square__eq,axiom,
    ! [R2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R2) ).

% real_divide_square_eq
tff(fact_913_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_914_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_915_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),one_one_complex) = A ).

% div_by_1
tff(fact_916_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_917_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_918_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_919_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_920_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_921_linorder__neqE__linordered__idom,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_922_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_923_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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)),E)),C) ).

% combine_common_factor
tff(fact_924_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E)),C) ).

% combine_common_factor
tff(fact_925_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_926_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).

% combine_common_factor
tff(fact_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_lambda__one,axiom,
    aTP_Lamm_af = aa_com3342990543769683525omplex(times_times_complex,one_one_complex) ).

% lambda_one
tff(fact_960_lambda__one,axiom,
    aTP_Lamm_ag = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_961_lambda__one,axiom,
    aTP_Lamm_ah = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_962_lambda__one,axiom,
    aTP_Lamm_ai = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_963_lambda__one,axiom,
    aTP_Lamm_aj = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_964_less__1__mult,axiom,
    ! [Mb: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_965_less__1__mult,axiom,
    ! [Mb: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Mb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_966_less__1__mult,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_967_less__1__mult,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_968_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real)) ) ).

% add_mono1
tff(fact_969_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_970_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_971_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_972_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)) ).

% less_add_one
tff(fact_973_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_974_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_975_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_976_add__le__add__imp__diff__le,axiom,
    ! [I2: real,K: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_977_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_978_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K: nat,Nb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_979_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_980_add__le__imp__le__diff,axiom,
    ! [I2: real,K: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_981_add__le__imp__le__diff,axiom,
    ! [I2: rat,K: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),aa_rat_rat(minus_minus_rat(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_982_add__le__imp__le__diff,axiom,
    ! [I2: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_983_add__le__imp__le__diff,axiom,
    ! [I2: int,K: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_984_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_985_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_986_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_987_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_988_square__diff__square__factored,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).

% square_diff_square_factored
tff(fact_989_square__diff__square__factored,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_rat_rat(minus_minus_rat(Xb),Y)) ).

% square_diff_square_factored
tff(fact_990_square__diff__square__factored,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) ).

% square_diff_square_factored
tff(fact_991_eq__add__iff2,axiom,
    ! [A: real,E: 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),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_992_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_993_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_994_eq__add__iff1,axiom,
    ! [A: real,E: 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),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),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)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_995_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_996_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_997_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_998_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_999_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1000_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1001_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1002_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1003_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_1004_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_1005_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_1006_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1007_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1008_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_1009_square__diff__one__factored,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb)),one_one_complex) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),one_one_complex)) ).

% square_diff_one_factored
tff(fact_1010_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_1011_square__diff__one__factored,axiom,
    ! [Xb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ).

% square_diff_one_factored
tff(fact_1012_square__diff__one__factored,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).

% square_diff_one_factored
tff(fact_1013_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
        some_nat(Mia),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_1014_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
        some_nat(Maa),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeLista,h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb),some_nat(Mia),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_1015_real__average__minus__first,axiom,
    ! [A: real,B: 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(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_1016_real__average__minus__second,axiom,
    ! [B: real,A: 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(plus_plus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_1017_field__less__half__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1018_field__less__half__sum,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1019_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Mia) ).

% vebt_mint.simps(3)
tff(fact_1020_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux,Uy,Uz)) = some_nat(Maa) ).

% vebt_maxt.simps(3)
tff(fact_1021_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ak(Ta),Xb)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_1022_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_al(Ta),Xb)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_1023_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(B,A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb3
tff(fact_1024_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb3
tff(fact_1025_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).

% max.absorb3
tff(fact_1026_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_1027_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb3
tff(fact_1028_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_1029_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_1030_max_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ).

% max.right_idem
tff(fact_1031_max_Oright__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ).

% max.right_idem
tff(fact_1032_max_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.right_idem
tff(fact_1033_max_Oright__idem,axiom,
    ! [A: code_integer,B: code_integer] : ord_max_Code_integer(ord_max_Code_integer(A,B),B) = ord_max_Code_integer(A,B) ).

% max.right_idem
tff(fact_1034_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ).

% max.left_idem
tff(fact_1035_max_Oleft__idem,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ).

% max.left_idem
tff(fact_1036_max_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.left_idem
tff(fact_1037_max_Oleft__idem,axiom,
    ! [A: code_integer,B: code_integer] : ord_max_Code_integer(A,ord_max_Code_integer(A,B)) = ord_max_Code_integer(A,B) ).

% max.left_idem
tff(fact_1038_max_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),A) = A ).

% max.idem
tff(fact_1039_max_Oidem,axiom,
    ! [A: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),A) = A ).

% max.idem
tff(fact_1040_max_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),A) = A ).

% max.idem
tff(fact_1041_max_Oidem,axiom,
    ! [A: code_integer] : ord_max_Code_integer(A,A) = A ).

% max.idem
tff(fact_1042_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_1043_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_1044_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),A)
    <=> ( ord_le2932123472753598470d_enat(B,A)
        & ord_le2932123472753598470d_enat(C,A) ) ) ).

% max.bounded_iff
tff(fact_1045_max_Obounded__iff,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A) ) ) ).

% max.bounded_iff
tff(fact_1046_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.bounded_iff
tff(fact_1047_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.bounded_iff
tff(fact_1048_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.bounded_iff
tff(fact_1049_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.bounded_iff
tff(fact_1050_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(A,B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb2
tff(fact_1051_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb2
tff(fact_1052_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_1053_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb2
tff(fact_1054_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_1055_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_1056_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb1
tff(fact_1057_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb1
tff(fact_1058_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_1059_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb1
tff(fact_1060_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_1061_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_1062_max__less__iff__conj,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y),Z)
    <=> ( ord_le72135733267957522d_enat(Xb,Z)
        & ord_le72135733267957522d_enat(Y,Z) ) ) ).

% max_less_iff_conj
tff(fact_1063_max__less__iff__conj,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(Xb,Y)),Z)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Z)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_1064_max__less__iff__conj,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_1065_max__less__iff__conj,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xb),Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_1066_max__less__iff__conj,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_1067_max__less__iff__conj,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_1068_max__less__iff__conj,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_1069_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le72135733267957522d_enat(A,B)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb4
tff(fact_1070_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb4
tff(fact_1071_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).

% max.absorb4
tff(fact_1072_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_1073_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb4
tff(fact_1074_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_1075_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_1076_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X: real] : member_real(X,S3)
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( member_real(X2,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z4) )
       => ? [Y4: real] :
            ( ! [X: real] :
                ( member_real(X,S3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y4) )
            & ! [Z4: real] :
                ( ! [X2: real] :
                    ( member_real(X2,S3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),Z4) ) ) ) ) ).

% complete_real
tff(fact_1077_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_1078_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) ) ).

% real_arch_pow
tff(fact_1079_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.left_commute
tff(fact_1080_max_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),C)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)) ).

% max.left_commute
tff(fact_1081_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),C)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.left_commute
tff(fact_1082_max_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : ord_max_Code_integer(B,ord_max_Code_integer(A,C)) = ord_max_Code_integer(A,ord_max_Code_integer(B,C)) ).

% max.left_commute
tff(fact_1083_max_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),A) ).

% max.commute
tff(fact_1084_max_Ocommute,axiom,
    ! [A: extended_enat,B: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),A) ).

% max.commute
tff(fact_1085_max_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,B),A) ).

% max.commute
tff(fact_1086_max_Ocommute,axiom,
    ! [A: code_integer,B: code_integer] : ord_max_Code_integer(A,B) = ord_max_Code_integer(B,A) ).

% max.commute
tff(fact_1087_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.assoc
tff(fact_1088_max_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)),C) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)) ).

% max.assoc
tff(fact_1089_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.assoc
tff(fact_1090_max_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : ord_max_Code_integer(ord_max_Code_integer(A,B),C) = ord_max_Code_integer(A,ord_max_Code_integer(B,C)) ).

% max.assoc
tff(fact_1091_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_am,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_1092_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_an(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_1093_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ao,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_1094_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_ap,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_1095_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_1096_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_1097_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_1098_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,B)
     => ord_le2932123472753598470d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.coboundedI2
tff(fact_1099_max_OcoboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),ord_max_Code_integer(A,B)) ) ).

% max.coboundedI2
tff(fact_1100_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI2
tff(fact_1101_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI2
tff(fact_1102_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_1103_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_1104_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,A)
     => ord_le2932123472753598470d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.coboundedI1
tff(fact_1105_max_OcoboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),ord_max_Code_integer(A,B)) ) ).

% max.coboundedI1
tff(fact_1106_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI1
tff(fact_1107_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.coboundedI1
tff(fact_1108_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_1109_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_1110_max_Oabsorb__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(A,B)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1111_max_Oabsorb__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
    <=> ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb_iff2
tff(fact_1112_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1113_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1114_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1115_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_1116_max_Oabsorb__iff1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
    <=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_1117_max_Oabsorb__iff1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
    <=> ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb_iff1
tff(fact_1118_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_1119_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_1120_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_1121_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_1122_le__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y))
    <=> ( ord_le2932123472753598470d_enat(Z,Xb)
        | ord_le2932123472753598470d_enat(Z,Y) ) ) ).

% le_max_iff_disj
tff(fact_1123_le__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),ord_max_Code_integer(Xb,Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_1124_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_1125_le__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_1126_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_1127_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_1128_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ord_le2932123472753598470d_enat(B,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ).

% max.cobounded2
tff(fact_1129_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),ord_max_Code_integer(A,B)) ).

% max.cobounded2
tff(fact_1130_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded2
tff(fact_1131_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded2
tff(fact_1132_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_1133_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_1134_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ord_le2932123472753598470d_enat(A,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ).

% max.cobounded1
tff(fact_1135_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),ord_max_Code_integer(A,B)) ).

% max.cobounded1
tff(fact_1136_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded1
tff(fact_1137_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ).

% max.cobounded1
tff(fact_1138_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_1139_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_1140_max_Oorder__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
    <=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.order_iff
tff(fact_1141_max_Oorder__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
    <=> ( A = ord_max_Code_integer(A,B) ) ) ).

% max.order_iff
tff(fact_1142_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_1143_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.order_iff
tff(fact_1144_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_1145_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_1146_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( ord_le2932123472753598470d_enat(C,A)
       => ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),A) ) ) ).

% max.boundedI
tff(fact_1147_max_OboundedI,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A) ) ) ).

% max.boundedI
tff(fact_1148_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A) ) ) ).

% max.boundedI
tff(fact_1149_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A) ) ) ).

% max.boundedI
tff(fact_1150_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_1151_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A) ) ) ).

% max.boundedI
tff(fact_1152_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),A)
     => ~ ( ord_le2932123472753598470d_enat(B,A)
         => ~ ord_le2932123472753598470d_enat(C,A) ) ) ).

% max.boundedE
tff(fact_1153_max_OboundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(B,C)),A)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A) ) ) ).

% max.boundedE
tff(fact_1154_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% max.boundedE
tff(fact_1155_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.boundedE
tff(fact_1156_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.boundedE
tff(fact_1157_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.boundedE
tff(fact_1158_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
     => ord_le2932123472753598470d_enat(B,A) ) ).

% max.orderI
tff(fact_1159_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = ord_max_Code_integer(A,B) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A) ) ).

% max.orderI
tff(fact_1160_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_1161_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% max.orderI
tff(fact_1162_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_1163_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_1164_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le2932123472753598470d_enat(B,A)
     => ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).

% max.orderE
tff(fact_1165_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),A)
     => ( A = ord_max_Code_integer(A,B) ) ) ).

% max.orderE
tff(fact_1166_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_1167_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).

% max.orderE
tff(fact_1168_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_1169_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_1170_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( ord_le2932123472753598470d_enat(C,A)
     => ( ord_le2932123472753598470d_enat(D,B)
       => ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,C),D),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ) ).

% max.mono
tff(fact_1171_max_Omono,axiom,
    ! [C: code_integer,A: code_integer,D: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(C,D)),ord_max_Code_integer(A,B)) ) ) ).

% max.mono
tff(fact_1172_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C),D)),aa_rat_rat(ord_max_rat(A),B)) ) ) ).

% max.mono
tff(fact_1173_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ) ).

% max.mono
tff(fact_1174_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_1175_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_1176_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(C,B)
     => ord_le72135733267957522d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_1177_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B)) ) ).

% max.strict_coboundedI2
tff(fact_1178_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_1179_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_1180_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_1181_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_1182_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_1183_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ord_le72135733267957522d_enat(C,A)
     => ord_le72135733267957522d_enat(C,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_1184_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(A,B)) ) ).

% max.strict_coboundedI1
tff(fact_1185_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_1186_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_1187_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_1188_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_1189_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_1190_max_Ostrict__order__iff,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(B,A)
    <=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_1191_max_Ostrict__order__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
    <=> ( ( A = ord_max_Code_integer(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_1192_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_1193_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_1194_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_1195_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_1196_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_1197_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C),A)
     => ~ ( ord_le72135733267957522d_enat(B,A)
         => ~ ord_le72135733267957522d_enat(C,A) ) ) ).

% max.strict_boundedE
tff(fact_1198_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(B,C)),A)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),A)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A) ) ) ).

% max.strict_boundedE
tff(fact_1199_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% max.strict_boundedE
tff(fact_1200_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),A)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_1201_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% max.strict_boundedE
tff(fact_1202_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_1203_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% max.strict_boundedE
tff(fact_1204_less__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( ord_le72135733267957522d_enat(Z,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y))
    <=> ( ord_le72135733267957522d_enat(Z,Xb)
        | ord_le72135733267957522d_enat(Z,Y) ) ) ).

% less_max_iff_disj
tff(fact_1205_less__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),ord_max_Code_integer(Xb,Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_1206_less__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_1207_less__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_1208_less__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_1209_less__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_1210_less__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_1211_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1212_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1213_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1214_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1215_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv: option_num] : vEBT_V819420779217536731ft_num(Uu,none_num,Uv) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1216_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1217_mult__diff__mult,axiom,
    ! [Xb: 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,Xb),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,Xb),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,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_1218_mult__diff__mult,axiom,
    ! [Xb: rat,Y: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Y)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),aa_rat_rat(minus_minus_rat(Y),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),A)),B)) ).

% mult_diff_mult
tff(fact_1219_mult__diff__mult,axiom,
    ! [Xb: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_1220_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1221_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1222_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1223_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
           => ( ( Xba = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A3) )
               => ! [B2: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B2) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1224_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xaa = some_num(V2)
           => ( ( Xba = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xaa = some_num(A3) )
               => ! [B2: num] :
                    ( ( Xba = some_num(B2) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1225_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
     => ( ( ( Xaa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xaa = some_nat(V2)
           => ( ( Xba = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xaa = some_nat(A3) )
               => ! [B2: nat] :
                    ( ( Xba = some_nat(B2) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A3),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1226_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X3) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_1227_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_1228_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1229_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1230_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_1231_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_1232_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_1233_field__sum__of__halves,axiom,
    ! [Xb: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_1234_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_1235_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_1236_max__bot2,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(Xb,bot_bot_set_nat) = Xb ).

% max_bot2
tff(fact_1237_max__bot2,axiom,
    ! [Xb: set_int] : ord_max_set_int(Xb,bot_bot_set_int) = Xb ).

% max_bot2
tff(fact_1238_max__bot2,axiom,
    ! [Xb: set_real] : ord_max_set_real(Xb,bot_bot_set_real) = Xb ).

% max_bot2
tff(fact_1239_max__bot2,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),bot_bot_nat) = Xb ).

% max_bot2
tff(fact_1240_max__bot2,axiom,
    ! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),bot_bo4199563552545308370d_enat) = Xb ).

% max_bot2
tff(fact_1241_max__bot,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xb) = Xb ).

% max_bot
tff(fact_1242_max__bot,axiom,
    ! [Xb: set_int] : ord_max_set_int(bot_bot_set_int,Xb) = Xb ).

% max_bot
tff(fact_1243_max__bot,axiom,
    ! [Xb: set_real] : ord_max_set_real(bot_bot_set_real,Xb) = Xb ).

% max_bot
tff(fact_1244_max__bot,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xb) = Xb ).

% max_bot
tff(fact_1245_max__bot,axiom,
    ! [Xb: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,bot_bo4199563552545308370d_enat),Xb) = Xb ).

% max_bot
tff(fact_1246_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_1247_empty__subsetI,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A2) ).

% empty_subsetI
tff(fact_1248_empty__subsetI,axiom,
    ! [A2: set_real] : ord_less_eq_set_real(bot_bot_set_real,A2) ).

% empty_subsetI
tff(fact_1249_empty__subsetI,axiom,
    ! [A2: set_int] : ord_less_eq_set_int(bot_bot_set_int,A2) ).

% empty_subsetI
tff(fact_1250_subset__empty,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1251_subset__empty,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,bot_bot_set_real)
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1252_subset__empty,axiom,
    ! [A2: set_int] :
      ( ord_less_eq_set_int(A2,bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1253_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( minus_minus_set_real(A2,B3) = bot_bot_set_real )
    <=> ord_less_eq_set_real(A2,B3) ) ).

% Diff_eq_empty_iff
tff(fact_1254_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat(A2,B3) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A2),B3) ) ).

% Diff_eq_empty_iff
tff(fact_1255_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( minus_minus_set_int(A2,B3) = bot_bot_set_int )
    <=> ord_less_eq_set_int(A2,B3) ) ).

% Diff_eq_empty_iff
tff(fact_1256_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_1257_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_1258_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_1259_order__refl,axiom,
    ! [Xb: set_int] : ord_less_eq_set_int(Xb,Xb) ).

% order_refl
tff(fact_1260_order__refl,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Xb) ).

% order_refl
tff(fact_1261_order__refl,axiom,
    ! [Xb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Xb) ).

% order_refl
tff(fact_1262_order__refl,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).

% order_refl
tff(fact_1263_order__refl,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).

% order_refl
tff(fact_1264_dual__order_Orefl,axiom,
    ! [A: set_int] : ord_less_eq_set_int(A,A) ).

% dual_order.refl
tff(fact_1265_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_1266_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_1267_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_1268_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_1269_subsetI,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => member_complex(X2,B3) )
     => ord_le211207098394363844omplex(A2,B3) ) ).

% subsetI
tff(fact_1270_subsetI,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => member_real(X2,B3) )
     => ord_less_eq_set_real(A2,B3) ) ).

% subsetI
tff(fact_1271_subsetI,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ! [X2: set_nat] :
          ( member_set_nat(X2,A2)
         => member_set_nat(X2,B3) )
     => ord_le6893508408891458716et_nat(A2,B3) ) ).

% subsetI
tff(fact_1272_subsetI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_set_nat_o(member_nat(X2),B3) )
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B3) ) ).

% subsetI
tff(fact_1273_subsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => member_int(X2,B3) )
     => ord_less_eq_set_int(A2,B3) ) ).

% subsetI
tff(fact_1274_psubsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ( A2 != B3 )
       => ord_less_set_int(A2,B3) ) ) ).

% psubsetI
tff(fact_1275_subset__antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,A2)
       => ( A2 = B3 ) ) ) ).

% subset_antisym
tff(fact_1276_Diff__idemp,axiom,
    ! [A2: set_nat,B3: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B3),B3) = minus_minus_set_nat(A2,B3) ).

% Diff_idemp
tff(fact_1277_Diff__iff,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A2,B3))
    <=> ( member_complex(C,A2)
        & ~ member_complex(C,B3) ) ) ).

% Diff_iff
tff(fact_1278_Diff__iff,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B3))
    <=> ( member_real(C,A2)
        & ~ member_real(C,B3) ) ) ).

% Diff_iff
tff(fact_1279_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( member_set_nat(C,minus_2163939370556025621et_nat(A2,B3))
    <=> ( member_set_nat(C,A2)
        & ~ member_set_nat(C,B3) ) ) ).

% Diff_iff
tff(fact_1280_Diff__iff,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A2,B3))
    <=> ( member_int(C,A2)
        & ~ member_int(C,B3) ) ) ).

% Diff_iff
tff(fact_1281_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B3))
    <=> ( aa_set_nat_o(member_nat(C),A2)
        & ~ aa_set_nat_o(member_nat(C),B3) ) ) ).

% Diff_iff
tff(fact_1282_DiffI,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( member_complex(C,A2)
     => ( ~ member_complex(C,B3)
       => member_complex(C,minus_811609699411566653omplex(A2,B3)) ) ) ).

% DiffI
tff(fact_1283_DiffI,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( member_real(C,A2)
     => ( ~ member_real(C,B3)
       => member_real(C,minus_minus_set_real(A2,B3)) ) ) ).

% DiffI
tff(fact_1284_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( member_set_nat(C,A2)
     => ( ~ member_set_nat(C,B3)
       => member_set_nat(C,minus_2163939370556025621et_nat(A2,B3)) ) ) ).

% DiffI
tff(fact_1285_DiffI,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( member_int(C,A2)
     => ( ~ member_int(C,B3)
       => member_int(C,minus_minus_set_int(A2,B3)) ) ) ).

% DiffI
tff(fact_1286_DiffI,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(C),A2)
     => ( ~ aa_set_nat_o(member_nat(C),B3)
       => aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B3)) ) ) ).

% DiffI
tff(fact_1287_Diff__empty,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).

% Diff_empty
tff(fact_1288_Diff__empty,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).

% Diff_empty
tff(fact_1289_Diff__empty,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).

% Diff_empty
tff(fact_1290_empty__Diff,axiom,
    ! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).

% empty_Diff
tff(fact_1291_empty__Diff,axiom,
    ! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).

% empty_Diff
tff(fact_1292_empty__Diff,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).

% empty_Diff
tff(fact_1293_Diff__cancel,axiom,
    ! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1294_Diff__cancel,axiom,
    ! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).

% Diff_cancel
tff(fact_1295_Diff__cancel,axiom,
    ! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1296_minus__set__def,axiom,
    ! [A2: set_real,B3: set_real] : minus_minus_set_real(A2,B3) = collect_real(minus_404528632214381484real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B3))) ).

% minus_set_def
tff(fact_1297_minus__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A2,B3) = collec213857154873943460nt_int(minus_827889751674334651_int_o(aTP_Lamm_aa(A2),aTP_Lamm_aa(B3))) ).

% minus_set_def
tff(fact_1298_minus__set__def,axiom,
    ! [A2: set_complex,B3: set_complex] : minus_811609699411566653omplex(A2,B3) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B3))) ).

% minus_set_def
tff(fact_1299_minus__set__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A2,B3) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B3))) ).

% minus_set_def
tff(fact_1300_minus__set__def,axiom,
    ! [A2: set_int,B3: set_int] : minus_minus_set_int(A2,B3) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B3))) ).

% minus_set_def
tff(fact_1301_minus__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] : minus_minus_set_nat(A2,B3) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B3))) ).

% minus_set_def
tff(fact_1302_set__diff__eq,axiom,
    ! [A2: set_real,B3: set_real] : minus_minus_set_real(A2,B3) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_aq(A2),B3)) ).

% set_diff_eq
tff(fact_1303_set__diff__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A2,B3) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_ar(A2),B3)) ).

% set_diff_eq
tff(fact_1304_set__diff__eq,axiom,
    ! [A2: set_complex,B3: set_complex] : minus_811609699411566653omplex(A2,B3) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_as(A2),B3)) ).

% set_diff_eq
tff(fact_1305_set__diff__eq,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A2,B3) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_at(A2),B3)) ).

% set_diff_eq
tff(fact_1306_set__diff__eq,axiom,
    ! [A2: set_int,B3: set_int] : minus_minus_set_int(A2,B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_au(A2),B3)) ).

% set_diff_eq
tff(fact_1307_set__diff__eq,axiom,
    ! [A2: set_nat,B3: set_nat] : minus_minus_set_nat(A2,B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_av(A2),B3)) ).

% set_diff_eq
tff(fact_1308_DiffD2,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A2,B3))
     => ~ member_complex(C,B3) ) ).

% DiffD2
tff(fact_1309_DiffD2,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B3))
     => ~ member_real(C,B3) ) ).

% DiffD2
tff(fact_1310_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( member_set_nat(C,minus_2163939370556025621et_nat(A2,B3))
     => ~ member_set_nat(C,B3) ) ).

% DiffD2
tff(fact_1311_DiffD2,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A2,B3))
     => ~ member_int(C,B3) ) ).

% DiffD2
tff(fact_1312_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B3))
     => ~ aa_set_nat_o(member_nat(C),B3) ) ).

% DiffD2
tff(fact_1313_DiffD1,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A2,B3))
     => member_complex(C,A2) ) ).

% DiffD1
tff(fact_1314_DiffD1,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B3))
     => member_real(C,A2) ) ).

% DiffD1
tff(fact_1315_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( member_set_nat(C,minus_2163939370556025621et_nat(A2,B3))
     => member_set_nat(C,A2) ) ).

% DiffD1
tff(fact_1316_DiffD1,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A2,B3))
     => member_int(C,A2) ) ).

% DiffD1
tff(fact_1317_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B3))
     => aa_set_nat_o(member_nat(C),A2) ) ).

% DiffD1
tff(fact_1318_DiffE,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A2,B3))
     => ~ ( member_complex(C,A2)
         => member_complex(C,B3) ) ) ).

% DiffE
tff(fact_1319_DiffE,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A2,B3))
     => ~ ( member_real(C,A2)
         => member_real(C,B3) ) ) ).

% DiffE
tff(fact_1320_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( member_set_nat(C,minus_2163939370556025621et_nat(A2,B3))
     => ~ ( member_set_nat(C,A2)
         => member_set_nat(C,B3) ) ) ).

% DiffE
tff(fact_1321_DiffE,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A2,B3))
     => ~ ( member_int(C,A2)
         => member_int(C,B3) ) ) ).

% DiffE
tff(fact_1322_DiffE,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(member_nat(C),minus_minus_set_nat(A2,B3))
     => ~ ( aa_set_nat_o(member_nat(C),A2)
         => aa_set_nat_o(member_nat(C),B3) ) ) ).

% DiffE
tff(fact_1323_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,Y)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1324_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_less_set_complex(A2,B3)
     => ? [B2: complex] : member_complex(B2,minus_811609699411566653omplex(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_1325_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ord_less_set_real(A2,B3)
     => ? [B2: real] : member_real(B2,minus_minus_set_real(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_1326_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ord_less_set_set_nat(A2,B3)
     => ? [B2: set_nat] : member_set_nat(B2,minus_2163939370556025621et_nat(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_1327_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
     => ? [B2: int] : member_int(B2,minus_minus_set_int(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_1328_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B3)
     => ? [B2: nat] : aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B3,A2)) ) ).

% psubset_imp_ex_mem
tff(fact_1329_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1330_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1331_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1332_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_1333_le__cases3,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1334_le__cases3,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1335_le__cases3,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1336_le__cases3,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1337_order__class_Oorder__eq__iff,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( Xb = Y )
    <=> ( ord_less_eq_set_int(Xb,Y)
        & ord_less_eq_set_int(Y,Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1338_order__class_Oorder__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1339_order__class_Oorder__eq__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1340_order__class_Oorder__eq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1341_order__class_Oorder__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_1342_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ord_less_eq_set_int(B,C)
       => ord_less_eq_set_int(A,C) ) ) ).

% ord_eq_le_trans
tff(fact_1343_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_1344_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_1345_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_1346_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_1347_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ( B = C )
       => ord_less_eq_set_int(A,C) ) ) ).

% ord_le_eq_trans
tff(fact_1348_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_1349_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_1350_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_1351_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_1352_order__antisym,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
     => ( ord_less_eq_set_int(Y,Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_1353_order__antisym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_1354_order__antisym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_1355_order__antisym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_1356_order__antisym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_1357_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ord_less_eq_set_int(B,C)
       => ord_less_eq_set_int(A,C) ) ) ).

% order.trans
tff(fact_1358_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% order.trans
tff(fact_1359_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% order.trans
tff(fact_1360_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% order.trans
tff(fact_1361_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% order.trans
tff(fact_1362_order__trans,axiom,
    ! [Xb: set_int,Y: set_int,Z: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
     => ( ord_less_eq_set_int(Y,Z)
       => ord_less_eq_set_int(Xb,Z) ) ) ).

% order_trans
tff(fact_1363_order__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) ) ) ).

% order_trans
tff(fact_1364_order__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) ) ) ).

% order_trans
tff(fact_1365_order__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) ) ) ).

% order_trans
tff(fact_1366_order__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) ) ) ).

% order_trans
tff(fact_1367_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
     => ( ! [A3: rat,B2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_1368_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B2)
         => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
     => ( ! [A3: num,B2: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B2),A3)
           => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
       => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_1369_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
     => ( ! [A3: nat,B2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_1370_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B2)
         => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
     => ( ! [A3: int,B2: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B2),A3)
           => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
       => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_1371_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( ord_less_eq_set_int(B,A)
        & ord_less_eq_set_int(A,B) ) ) ).

% dual_order.eq_iff
tff(fact_1372_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1373_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1374_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1375_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_1376_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_eq_set_int(B,A)
     => ( ord_less_eq_set_int(A,B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1377_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1378_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1379_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1380_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_1381_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ord_less_eq_set_int(B,A)
     => ( ord_less_eq_set_int(C,B)
       => ord_less_eq_set_int(C,A) ) ) ).

% dual_order.trans
tff(fact_1382_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% dual_order.trans
tff(fact_1383_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% dual_order.trans
tff(fact_1384_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_1385_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% dual_order.trans
tff(fact_1386_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ord_less_eq_set_int(B,A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1387_antisym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1388_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1389_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1390_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_1391_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( ord_less_eq_set_int(A,B)
        & ord_less_eq_set_int(B,A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1392_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1393_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1394_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1395_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_1396_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1397_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1398_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1399_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1400_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1401_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1402_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1403_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X2)),aa_int_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_1404_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1405_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_1406_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1407_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1408_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1409_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1410_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1411_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1412_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1413_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1414_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1415_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_1416_order__eq__refl,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( Xb = Y )
     => ord_less_eq_set_int(Xb,Y) ) ).

% order_eq_refl
tff(fact_1417_order__eq__refl,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% order_eq_refl
tff(fact_1418_order__eq__refl,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% order_eq_refl
tff(fact_1419_order__eq__refl,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_eq_refl
tff(fact_1420_order__eq__refl,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_eq_refl
tff(fact_1421_linorder__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_linear
tff(fact_1422_linorder__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_linear
tff(fact_1423_linorder__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_linear
tff(fact_1424_linorder__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_linear
tff(fact_1425_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1426_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1427_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1428_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1429_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1430_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1431_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1432_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1433_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1434_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1435_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1436_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1437_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1438_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1439_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1440_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1441_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1442_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1443_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1444_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X2)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1445_linorder__le__cases,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_le_cases
tff(fact_1446_linorder__le__cases,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_le_cases
tff(fact_1447_linorder__le__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_le_cases
tff(fact_1448_linorder__le__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_le_cases
tff(fact_1449_order__antisym__conv,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( ord_less_eq_set_int(Y,Xb)
     => ( ord_less_eq_set_int(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_1450_order__antisym__conv,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_1451_order__antisym__conv,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_1452_order__antisym__conv,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_1453_order__antisym__conv,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_1454_lt__ex,axiom,
    ! [Xb: real] :
    ? [Y4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),Xb) ).

% lt_ex
tff(fact_1455_lt__ex,axiom,
    ! [Xb: rat] :
    ? [Y4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y4),Xb) ).

% lt_ex
tff(fact_1456_lt__ex,axiom,
    ! [Xb: int] :
    ? [Y4: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y4),Xb) ).

% lt_ex
tff(fact_1457_gt__ex,axiom,
    ! [Xb: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_1) ).

% gt_ex
tff(fact_1458_gt__ex,axiom,
    ! [Xb: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),X_1) ).

% gt_ex
tff(fact_1459_gt__ex,axiom,
    ! [Xb: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_1) ).

% gt_ex
tff(fact_1460_gt__ex,axiom,
    ! [Xb: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_1) ).

% gt_ex
tff(fact_1461_dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Y) ) ) ).

% dense
tff(fact_1462_dense,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ? [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z3)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),Y) ) ) ).

% dense
tff(fact_1463_less__imp__neq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1464_less__imp__neq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1465_less__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1466_less__imp__neq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1467_less__imp__neq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_1468_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order.asym
tff(fact_1469_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_1470_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_1471_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_1472_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_1473_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1474_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1475_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1476_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1477_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_1478_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1479_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1480_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1481_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1482_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_1483_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X2: nat] :
          ( ! [Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X2)
             => aa_nat_o(P,Y2) )
         => aa_nat_o(P,X2) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_1484_antisym__conv3,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1485_antisym__conv3,axiom,
    ! [Y: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1486_antisym__conv3,axiom,
    ! [Y: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1487_antisym__conv3,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1488_antisym__conv3,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_1489_linorder__cases,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ( Xb != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1490_linorder__cases,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( ( Xb != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1491_linorder__cases,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( ( Xb != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1492_linorder__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( ( Xb != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1493_linorder__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( ( Xb != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_cases
tff(fact_1494_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% dual_order.asym
tff(fact_1495_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_1496_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_1497_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_1498_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_1499_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_1500_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_1501_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_1502_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_1503_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_1504_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
             => ~ aa_nat_o(P,M5) ) ) ) ).

% exists_least_iff
tff(fact_1505_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A3: real,B2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B2)
         => aa_real_o(aa_real_fun_real_o(P,A3),B2) )
     => ( ! [A3: real] : aa_real_o(aa_real_fun_real_o(P,A3),A3)
       => ( ! [A3: real,B2: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B2),A3)
             => aa_real_o(aa_real_fun_real_o(P,A3),B2) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1506_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A3),A3)
       => ( ! [A3: rat,B2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1507_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B2)
         => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(P,A3),A3)
       => ( ! [A3: num,B2: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B2),A3)
             => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
         => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1508_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),A3)
       => ( ! [A3: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1509_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B2)
         => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P,A3),A3)
       => ( ! [A3: int,B2: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B2),A3)
             => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
         => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_1510_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans
tff(fact_1511_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_1512_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans
tff(fact_1513_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_1514_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_1515_not__less__iff__gr__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1516_not__less__iff__gr__or__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1517_not__less__iff__gr__or__eq,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1518_not__less__iff__gr__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1519_not__less__iff__gr__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_1520_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1521_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1522_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1523_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1524_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_1525_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1526_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1527_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1528_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1529_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_1530_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1531_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1532_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1533_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1534_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_1535_linorder__neqE,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_1536_linorder__neqE,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_1537_linorder__neqE,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_1538_linorder__neqE,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_1539_linorder__neqE,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_1540_order__less__asym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_asym
tff(fact_1541_order__less__asym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_asym
tff(fact_1542_order__less__asym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_asym
tff(fact_1543_order__less__asym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_asym
tff(fact_1544_order__less__asym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_asym
tff(fact_1545_linorder__neq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1546_linorder__neq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1547_linorder__neq__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1548_linorder__neq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1549_linorder__neq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_1550_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order_less_asym'
tff(fact_1551_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_1552_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_1553_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_1554_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_1555_order__less__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1556_order__less__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1557_order__less__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1558_order__less__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1559_order__less__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_trans
tff(fact_1560_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1561_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1562_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1563_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1564_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1565_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1566_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1567_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1568_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1569_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_1570_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1571_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1572_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1573_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1574_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1575_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1576_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1577_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1578_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1579_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_1580_order__less__irrefl,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).

% order_less_irrefl
tff(fact_1581_order__less__irrefl,axiom,
    ! [Xb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Xb) ).

% order_less_irrefl
tff(fact_1582_order__less__irrefl,axiom,
    ! [Xb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Xb) ).

% order_less_irrefl
tff(fact_1583_order__less__irrefl,axiom,
    ! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).

% order_less_irrefl
tff(fact_1584_order__less__irrefl,axiom,
    ! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).

% order_less_irrefl
tff(fact_1585_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1586_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1587_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1588_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1589_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1590_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1591_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1592_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1593_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1594_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_1595_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1596_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1597_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X2)),aa_real_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1598_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1599_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X2)),aa_real_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1600_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1601_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1602_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1603_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1604_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_1605_order__less__not__sym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_not_sym
tff(fact_1606_order__less__not__sym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_1607_order__less__not__sym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_not_sym
tff(fact_1608_order__less__not__sym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_1609_order__less__not__sym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_not_sym
tff(fact_1610_order__less__imp__triv,axiom,
    ! [Xb: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1611_order__less__imp__triv,axiom,
    ! [Xb: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1612_order__less__imp__triv,axiom,
    ! [Xb: num,Y: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1613_order__less__imp__triv,axiom,
    ! [Xb: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1614_order__less__imp__triv,axiom,
    ! [Xb: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_1615_linorder__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      | ( Xb = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_less_linear
tff(fact_1616_linorder__less__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      | ( Xb = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_1617_linorder__less__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      | ( Xb = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_less_linear
tff(fact_1618_linorder__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      | ( Xb = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_1619_linorder__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      | ( Xb = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_less_linear
tff(fact_1620_order__less__imp__not__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1621_order__less__imp__not__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1622_order__less__imp__not__eq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1623_order__less__imp__not__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1624_order__less__imp__not__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_1625_order__less__imp__not__eq2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1626_order__less__imp__not__eq2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1627_order__less__imp__not__eq2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1628_order__less__imp__not__eq2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1629_order__less__imp__not__eq2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_1630_order__less__imp__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_1631_order__less__imp__not__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_1632_order__less__imp__not__less,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_1633_order__less__imp__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_1634_order__less__imp__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_1635_Diff__mono,axiom,
    ! [A2: set_nat,C2: set_nat,D2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D2),B3)
       => aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,B3)),minus_minus_set_nat(C2,D2)) ) ) ).

% Diff_mono
tff(fact_1636_Diff__mono,axiom,
    ! [A2: set_int,C2: set_int,D2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,C2)
     => ( ord_less_eq_set_int(D2,B3)
       => ord_less_eq_set_int(minus_minus_set_int(A2,B3),minus_minus_set_int(C2,D2)) ) ) ).

% Diff_mono
tff(fact_1637_Diff__subset,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A2,B3)),A2) ).

% Diff_subset
tff(fact_1638_Diff__subset,axiom,
    ! [A2: set_int,B3: set_int] : ord_less_eq_set_int(minus_minus_set_int(A2,B3),A2) ).

% Diff_subset
tff(fact_1639_double__diff,axiom,
    ! [A2: set_nat,B3: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C2)
       => ( minus_minus_set_nat(B3,minus_minus_set_nat(C2,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_1640_double__diff,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,C2)
       => ( minus_minus_set_int(B3,minus_minus_set_int(C2,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_1641_in__mono,axiom,
    ! [A2: set_complex,B3: set_complex,Xb: complex] :
      ( ord_le211207098394363844omplex(A2,B3)
     => ( member_complex(Xb,A2)
       => member_complex(Xb,B3) ) ) ).

% in_mono
tff(fact_1642_in__mono,axiom,
    ! [A2: set_real,B3: set_real,Xb: real] :
      ( ord_less_eq_set_real(A2,B3)
     => ( member_real(Xb,A2)
       => member_real(Xb,B3) ) ) ).

% in_mono
tff(fact_1643_in__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,Xb: set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
     => ( member_set_nat(Xb,A2)
       => member_set_nat(Xb,B3) ) ) ).

% in_mono
tff(fact_1644_in__mono,axiom,
    ! [A2: set_nat,B3: set_nat,Xb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( aa_set_nat_o(member_nat(Xb),A2)
       => aa_set_nat_o(member_nat(Xb),B3) ) ) ).

% in_mono
tff(fact_1645_in__mono,axiom,
    ! [A2: set_int,B3: set_int,Xb: int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( member_int(Xb,A2)
       => member_int(Xb,B3) ) ) ).

% in_mono
tff(fact_1646_subsetD,axiom,
    ! [A2: set_complex,B3: set_complex,C: complex] :
      ( ord_le211207098394363844omplex(A2,B3)
     => ( member_complex(C,A2)
       => member_complex(C,B3) ) ) ).

% subsetD
tff(fact_1647_subsetD,axiom,
    ! [A2: set_real,B3: set_real,C: real] :
      ( ord_less_eq_set_real(A2,B3)
     => ( member_real(C,A2)
       => member_real(C,B3) ) ) ).

% subsetD
tff(fact_1648_subsetD,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
     => ( member_set_nat(C,A2)
       => member_set_nat(C,B3) ) ) ).

% subsetD
tff(fact_1649_subsetD,axiom,
    ! [A2: set_nat,B3: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B3) ) ) ).

% subsetD
tff(fact_1650_subsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( member_int(C,A2)
       => member_int(C,B3) ) ) ).

% subsetD
tff(fact_1651_psubsetE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
     => ~ ( ord_less_eq_set_int(A2,B3)
         => ord_less_eq_set_int(B3,A2) ) ) ).

% psubsetE
tff(fact_1652_equalityE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ~ ( ord_less_eq_set_int(A2,B3)
         => ~ ord_less_eq_set_int(B3,A2) ) ) ).

% equalityE
tff(fact_1653_subset__eq,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(A2,B3)
    <=> ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_complex(X3,B3) ) ) ).

% subset_eq
tff(fact_1654_subset__eq,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ord_less_eq_set_real(A2,B3)
    <=> ! [X3: real] :
          ( member_real(X3,A2)
         => member_real(X3,B3) ) ) ).

% subset_eq
tff(fact_1655_subset__eq,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
    <=> ! [X3: set_nat] :
          ( member_set_nat(X3,A2)
         => member_set_nat(X3,B3) ) ) ).

% subset_eq
tff(fact_1656_subset__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_set_nat_o(member_nat(X3),B3) ) ) ).

% subset_eq
tff(fact_1657_subset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ! [X3: int] :
          ( member_int(X3,A2)
         => member_int(X3,B3) ) ) ).

% subset_eq
tff(fact_1658_equalityD1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ord_less_eq_set_int(A2,B3) ) ).

% equalityD1
tff(fact_1659_equalityD2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ord_less_eq_set_int(B3,A2) ) ).

% equalityD2
tff(fact_1660_psubset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
    <=> ( ord_less_eq_set_int(A2,B3)
        & ( A2 != B3 ) ) ) ).

% psubset_eq
tff(fact_1661_subset__iff,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(A2,B3)
    <=> ! [T: complex] :
          ( member_complex(T,A2)
         => member_complex(T,B3) ) ) ).

% subset_iff
tff(fact_1662_subset__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ord_less_eq_set_real(A2,B3)
    <=> ! [T: real] :
          ( member_real(T,A2)
         => member_real(T,B3) ) ) ).

% subset_iff
tff(fact_1663_subset__iff,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
    <=> ! [T: set_nat] :
          ( member_set_nat(T,A2)
         => member_set_nat(T,B3) ) ) ).

% subset_iff
tff(fact_1664_subset__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A2)
         => aa_set_nat_o(member_nat(T),B3) ) ) ).

% subset_iff
tff(fact_1665_subset__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ! [T: int] :
          ( member_int(T,A2)
         => member_int(T,B3) ) ) ).

% subset_iff
tff(fact_1666_subset__refl,axiom,
    ! [A2: set_int] : ord_less_eq_set_int(A2,A2) ).

% subset_refl
tff(fact_1667_Collect__mono,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X2)
         => aa_Pro4592068669816069595_int_o(Q,X2) )
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q)) ) ).

% Collect_mono
tff(fact_1668_Collect__mono,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X2: complex] :
          ( aa_complex_o(P,X2)
         => aa_complex_o(Q,X2) )
     => ord_le211207098394363844omplex(collect_complex(P),collect_complex(Q)) ) ).

% Collect_mono
tff(fact_1669_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
         => aa_set_nat_o(Q,X2) )
     => ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_1670_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
         => aa_nat_o(Q,X2) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_1671_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
         => aa_int_o(Q,X2) )
     => ord_less_eq_set_int(collect_int(P),collect_int(Q)) ) ).

% Collect_mono
tff(fact_1672_subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,C2)
       => ord_less_eq_set_int(A2,C2) ) ) ).

% subset_trans
tff(fact_1673_set__eq__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
    <=> ( ord_less_eq_set_int(A2,B3)
        & ord_less_eq_set_int(B3,A2) ) ) ).

% set_eq_subset
tff(fact_1674_Collect__mono__iff,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q))
    <=> ! [X3: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X3)
         => aa_Pro4592068669816069595_int_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1675_Collect__mono__iff,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ord_le211207098394363844omplex(collect_complex(P),collect_complex(Q))
    <=> ! [X3: complex] :
          ( aa_complex_o(P,X3)
         => aa_complex_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1676_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q))
    <=> ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
         => aa_set_nat_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1677_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q))
    <=> ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1678_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ord_less_eq_set_int(collect_int(P),collect_int(Q))
    <=> ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_1679_psubset__imp__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
     => ord_less_eq_set_int(A2,B3) ) ).

% psubset_imp_subset
tff(fact_1680_psubset__subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( ord_less_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,C2)
       => ord_less_set_int(A2,C2) ) ) ).

% psubset_subset_trans
tff(fact_1681_subset__not__subset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
    <=> ( ord_less_eq_set_int(A2,B3)
        & ~ ord_less_eq_set_int(B3,A2) ) ) ).

% subset_not_subset_eq
tff(fact_1682_subset__psubset__trans,axiom,
    ! [A2: set_int,B3: set_int,C2: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_set_int(B3,C2)
       => ord_less_set_int(A2,C2) ) ) ).

% subset_psubset_trans
tff(fact_1683_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ( ord_less_set_int(A2,B3)
        | ( A2 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1684_empty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_aw) ).

% empty_def
tff(fact_1685_empty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_ax) ).

% empty_def
tff(fact_1686_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_ay) ).

% empty_def
tff(fact_1687_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_az) ).

% empty_def
tff(fact_1688_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_ba) ).

% empty_def
tff(fact_1689_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_bb) ).

% empty_def
tff(fact_1690_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_o] : ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A2),P)),A2) ).

% Collect_subset
tff(fact_1691_Collect__subset,axiom,
    ! [A2: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bd(A2),P)),A2) ).

% Collect_subset
tff(fact_1692_Collect__subset,axiom,
    ! [A2: set_complex,P: fun_complex_o] : ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A2),P)),A2) ).

% Collect_subset
tff(fact_1693_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bf(A2),P)),A2) ).

% Collect_subset
tff(fact_1694_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(A2),P))),A2) ).

% Collect_subset
tff(fact_1695_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_o] : ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A2),P)),A2) ).

% Collect_subset
tff(fact_1696_less__eq__set__def,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(A2,B3)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B3)) ) ).

% less_eq_set_def
tff(fact_1697_less__eq__set__def,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ord_less_eq_set_real(A2,B3)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_a(A2),aTP_Lamm_a(B3)) ) ).

% less_eq_set_def
tff(fact_1698_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B3)) ) ).

% less_eq_set_def
tff(fact_1699_less__eq__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B3)) ) ).

% less_eq_set_def
tff(fact_1700_less__eq__set__def,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B3)) ) ).

% less_eq_set_def
tff(fact_1701_leD,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% leD
tff(fact_1702_leD,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( ord_less_eq_set_int(Y,Xb)
     => ~ ord_less_set_int(Xb,Y) ) ).

% leD
tff(fact_1703_leD,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% leD
tff(fact_1704_leD,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).

% leD
tff(fact_1705_leD,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% leD
tff(fact_1706_leD,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% leD
tff(fact_1707_leI,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% leI
tff(fact_1708_leI,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% leI
tff(fact_1709_leI,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% leI
tff(fact_1710_leI,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% leI
tff(fact_1711_leI,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% leI
tff(fact_1712_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1713_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ ord_less_set_int(A,B)
    <=> ( ~ ord_less_eq_set_int(A,B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1714_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1715_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1716_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1717_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_1718_antisym__conv1,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1719_antisym__conv1,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ~ ord_less_set_int(Xb,Y)
     => ( ord_less_eq_set_int(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1720_antisym__conv1,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1721_antisym__conv1,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1722_antisym__conv1,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1723_antisym__conv1,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_1724_antisym__conv2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1725_antisym__conv2,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
     => ( ~ ord_less_set_int(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1726_antisym__conv2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1727_antisym__conv2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1728_antisym__conv2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1729_antisym__conv2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_1730_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X2) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_1731_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_1732_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_1733_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_1734_less__le__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1735_less__le__not__le,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_set_int(Xb,Y)
    <=> ( ord_less_eq_set_int(Xb,Y)
        & ~ ord_less_eq_set_int(Y,Xb) ) ) ).

% less_le_not_le
tff(fact_1736_less__le__not__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1737_less__le__not__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1738_less__le__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1739_less__le__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_1740_not__le__imp__less,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1741_not__le__imp__less,axiom,
    ! [Y: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1742_not__le__imp__less,axiom,
    ! [Y: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1743_not__le__imp__less,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1744_not__le__imp__less,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% not_le_imp_less
tff(fact_1745_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1746_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
    <=> ( ord_less_set_int(A,B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1747_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1748_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1749_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1750_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_1751_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1752_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(A,B)
    <=> ( ord_less_eq_set_int(A,B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1753_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1754_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1755_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1756_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_1757_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans1
tff(fact_1758_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ord_less_set_int(B,C)
       => ord_less_set_int(A,C) ) ) ).

% order.strict_trans1
tff(fact_1759_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_1760_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans1
tff(fact_1761_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_1762_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_1763_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans2
tff(fact_1764_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_set_int(A,B)
     => ( ord_less_eq_set_int(B,C)
       => ord_less_set_int(A,C) ) ) ).

% order.strict_trans2
tff(fact_1765_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_1766_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans2
tff(fact_1767_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_1768_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_1769_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1770_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(A,B)
    <=> ( ord_less_eq_set_int(A,B)
        & ~ ord_less_eq_set_int(B,A) ) ) ).

% order.strict_iff_not
tff(fact_1771_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1772_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1773_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1774_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_1775_dense__ge__bounded,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Xb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_1776_dense__ge__bounded,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Xb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_1777_dense__le__bounded,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_1778_dense__le__bounded,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_1779_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1780_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_eq_set_int(B,A)
    <=> ( ord_less_set_int(B,A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1781_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1782_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1783_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1784_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1785_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1786_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_set_int(B,A)
    <=> ( ord_less_eq_set_int(B,A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1787_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1788_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1789_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1790_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1791_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1792_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ord_less_eq_set_int(B,A)
     => ( ord_less_set_int(C,B)
       => ord_less_set_int(C,A) ) ) ).

% dual_order.strict_trans1
tff(fact_1793_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1794_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1795_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1796_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_1797_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1798_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ord_less_set_int(B,A)
     => ( ord_less_eq_set_int(C,B)
       => ord_less_set_int(C,A) ) ) ).

% dual_order.strict_trans2
tff(fact_1799_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1800_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1801_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1802_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_1803_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1804_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_set_int(B,A)
    <=> ( ord_less_eq_set_int(B,A)
        & ~ ord_less_eq_set_int(A,B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1805_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1806_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1807_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1808_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_1809_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_1810_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(A,B)
     => ord_less_eq_set_int(A,B) ) ).

% order.strict_implies_order
tff(fact_1811_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_1812_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_1813_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_1814_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_1815_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1816_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_set_int(B,A)
     => ord_less_eq_set_int(B,A) ) ).

% dual_order.strict_implies_order
tff(fact_1817_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1818_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1819_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1820_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_1821_order__le__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1822_order__le__less,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
    <=> ( ord_less_set_int(Xb,Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1823_order__le__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1824_order__le__less,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1825_order__le__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1826_order__le__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_1827_order__less__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1828_order__less__le,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_set_int(Xb,Y)
    <=> ( ord_less_eq_set_int(Xb,Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1829_order__less__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1830_order__less__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1831_order__less__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1832_order__less__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_1833_linorder__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_not_le
tff(fact_1834_linorder__not__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_not_le
tff(fact_1835_linorder__not__le,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_not_le
tff(fact_1836_linorder__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_not_le
tff(fact_1837_linorder__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_not_le
tff(fact_1838_linorder__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% linorder_not_less
tff(fact_1839_linorder__not__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_not_less
tff(fact_1840_linorder__not__less,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_not_less
tff(fact_1841_linorder__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_not_less
tff(fact_1842_linorder__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_not_less
tff(fact_1843_order__less__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1844_order__less__imp__le,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_set_int(Xb,Y)
     => ord_less_eq_set_int(Xb,Y) ) ).

% order_less_imp_le
tff(fact_1845_order__less__imp__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1846_order__less__imp__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1847_order__less__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1848_order__less__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_less_imp_le
tff(fact_1849_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1850_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ( A != B )
       => ord_less_set_int(A,B) ) ) ).

% order_le_neq_trans
tff(fact_1851_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1852_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1853_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1854_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_1855_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1856_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( ord_less_eq_set_int(A,B)
       => ord_less_set_int(A,B) ) ) ).

% order_neq_le_trans
tff(fact_1857_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1858_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1859_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1860_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_1861_order__le__less__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1862_order__le__less__trans,axiom,
    ! [Xb: set_int,Y: set_int,Z: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
     => ( ord_less_set_int(Y,Z)
       => ord_less_set_int(Xb,Z) ) ) ).

% order_le_less_trans
tff(fact_1863_order__le__less__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1864_order__le__less__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1865_order__le__less__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1866_order__le__less__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_1867_order__less__le__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1868_order__less__le__trans,axiom,
    ! [Xb: set_int,Y: set_int,Z: set_int] :
      ( ord_less_set_int(Xb,Y)
     => ( ord_less_eq_set_int(Y,Z)
       => ord_less_set_int(Xb,Z) ) ) ).

% order_less_le_trans
tff(fact_1869_order__less__le__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1870_order__less__le__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1871_order__less__le__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1872_order__less__le__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_1873_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1874_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1875_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1876_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1877_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1878_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1879_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1880_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1881_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1882_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_1883_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1884_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1885_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1886_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1887_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1888_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X2)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1889_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1890_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1891_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1892_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_1893_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1894_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1895_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X2)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1896_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X2)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1897_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X2)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1898_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X2)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1899_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1900_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X2)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1901_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X2)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1902_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X2)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_1903_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1904_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X2)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1905_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X2)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1906_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X2)),aa_nat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1907_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1908_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C)
       => ( ! [X2: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1909_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X2: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X2)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1910_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X2: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X2)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1911_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1912_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
       => ( ! [X2: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_1913_linorder__le__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1914_linorder__le__less__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1915_linorder__le__less__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1916_linorder__le__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1917_linorder__le__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_1918_order__le__imp__less__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1919_order__le__imp__less__or__eq,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
     => ( ord_less_set_int(Xb,Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1920_order__le__imp__less__or__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1921_order__le__imp__less__or__eq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1922_order__le__imp__less__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1923_order__le__imp__less__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1924_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_1925_bot_Oextremum,axiom,
    ! [A: set_real] : ord_less_eq_set_real(bot_bot_set_real,A) ).

% bot.extremum
tff(fact_1926_bot_Oextremum,axiom,
    ! [A: set_int] : ord_less_eq_set_int(bot_bot_set_int,A) ).

% bot.extremum
tff(fact_1927_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_1928_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_1929_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_1930_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ord_less_eq_set_int(A,bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_1931_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_1932_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1933_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_1934_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ord_less_eq_set_int(A,bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_1935_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_1936_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_1937_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ ord_less_set_int(A,bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_1938_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ ord_less_set_real(A,bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_1939_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_1940_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1941_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> ord_less_set_int(bot_bot_set_int,A) ) ).

% bot.not_eq_extremum
tff(fact_1942_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> ord_less_set_real(bot_bot_set_real,A) ) ).

% bot.not_eq_extremum
tff(fact_1943_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_1944_max__def,axiom,
    ! [A: extended_enat,B: extended_enat] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = $ite(ord_le2932123472753598470d_enat(A,B),B,A) ).

% max_def
tff(fact_1945_max__def,axiom,
    ! [A: code_integer,B: code_integer] :
      ord_max_Code_integer(A,B) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B),B,A) ).

% max_def
tff(fact_1946_max__def,axiom,
    ! [A: set_int,B: set_int] :
      ord_max_set_int(A,B) = $ite(ord_less_eq_set_int(A,B),B,A) ).

% max_def
tff(fact_1947_max__def,axiom,
    ! [A: rat,B: rat] :
      aa_rat_rat(ord_max_rat(A),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B),B,A) ).

% max_def
tff(fact_1948_max__def,axiom,
    ! [A: num,B: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B),B,A) ).

% max_def
tff(fact_1949_max__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).

% max_def
tff(fact_1950_max__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).

% max_def
tff(fact_1951_max__absorb1,axiom,
    ! [Y: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Y,Xb)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1952_max__absorb1,axiom,
    ! [Y: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xb)
     => ( ord_max_Code_integer(Xb,Y) = Xb ) ) ).

% max_absorb1
tff(fact_1953_max__absorb1,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( ord_less_eq_set_int(Y,Xb)
     => ( ord_max_set_int(Xb,Y) = Xb ) ) ).

% max_absorb1
tff(fact_1954_max__absorb1,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ( aa_rat_rat(ord_max_rat(Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1955_max__absorb1,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1956_max__absorb1,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1957_max__absorb1,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_1958_max__absorb2,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ord_le2932123472753598470d_enat(Xb,Y)
     => ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1959_max__absorb2,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Y)
     => ( ord_max_Code_integer(Xb,Y) = Y ) ) ).

% max_absorb2
tff(fact_1960_max__absorb2,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
     => ( ord_max_set_int(Xb,Y) = Y ) ) ).

% max_absorb2
tff(fact_1961_max__absorb2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_rat(ord_max_rat(Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1962_max__absorb2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1963_max__absorb2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1964_max__absorb2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_1965_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_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),C) ).

% times_divide_eq_right
tff(fact_1988_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_1989_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_1990_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),B) ).

% divide_divide_eq_right
tff(fact_1991_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_1992_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_1993_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) ).

% divide_divide_eq_left
tff(fact_1994_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_1995_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_1996_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)),A) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),A)),C) ).

% times_divide_eq_left
tff(fact_1997_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_1998_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_1999_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_2000_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_2001_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_2002_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_2003_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_2004_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_2005_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_2006_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_2007_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_2008_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_2009_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_2010_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_2011_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_2012_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_2013_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_2014_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_2015_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_2016_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_2017_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_2018_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_2019_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_2020_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_2021_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_2022_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_2023_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_2024_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_2025_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_2026_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_2027_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_2028_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,one_one_complex),A) = A ).

% mult_1
tff(fact_2029_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_2030_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_2031_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_2032_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_2033_linordered__field__no__ub,axiom,
    ! [X: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),X_1) ).

% linordered_field_no_ub
tff(fact_2034_linordered__field__no__ub,axiom,
    ! [X: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_1) ).

% linordered_field_no_ub
tff(fact_2035_linordered__field__no__lb,axiom,
    ! [X: real] :
    ? [Y4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),X) ).

% linordered_field_no_lb
tff(fact_2036_linordered__field__no__lb,axiom,
    ! [X: rat] :
    ? [Y4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y4),X) ).

% linordered_field_no_lb
tff(fact_2037_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_2038_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_2039_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_2040_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_2041_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_2042_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_2043_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_2044_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_2045_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_2046_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_2047_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_2048_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_2049_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_2050_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_2051_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_2052_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_2053_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_2054_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_2055_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_2056_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_2057_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2058_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I2),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2059_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2060_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2061_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_2062_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_2063_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_2064_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_2065_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B3) = 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_2066_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B3) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_2067_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3 = 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),B3) = 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_2068_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = 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),B3) = 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_2069_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_2070_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_2071_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_2072_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_2073_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_2074_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_2075_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_2076_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_2077_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_2078_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_2079_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_2080_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_2081_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_2082_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_2083_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_2084_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_2085_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_2086_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_2087_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_2088_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_2089_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_2090_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_2091_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_2092_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_2093_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_2094_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_2095_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_2096_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_2097_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_2098_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_2099_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_2100_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_2101_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_2102_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_2103_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_2104_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_2105_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_2106_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_2107_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2108_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2109_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2110_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2111_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( I2 = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2112_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2113_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2114_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2115_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2116_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2117_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2118_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2119_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_mono
tff(fact_2120_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_2121_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_2122_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_2123_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_left_mono
tff(fact_2124_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_left_mono
tff(fact_2125_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_2126_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_2127_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_2128_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_right_mono
tff(fact_2129_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_right_mono
tff(fact_2130_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_2131_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_2132_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_2133_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_2134_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2135_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2136_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_2137_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_2138_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2139_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2140_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_2141_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2142_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2143_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2144_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2145_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( I2 = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2146_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2147_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2148_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2149_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I2),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2150_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2151_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2152_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2153_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_strict_mono
tff(fact_2154_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_2155_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_2156_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_2157_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% add_strict_left_mono
tff(fact_2158_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_strict_left_mono
tff(fact_2159_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_2160_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_2161_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ).

% add_strict_right_mono
tff(fact_2162_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_strict_right_mono
tff(fact_2163_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_2164_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_2165_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_2166_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_2167_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_2168_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_2169_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_2170_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_2171_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_2172_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_2173_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2174_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2175_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2176_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_right_mono
tff(fact_2177_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_right_mono
tff(fact_2178_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_2179_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_left_mono
tff(fact_2180_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_left_mono
tff(fact_2181_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_2182_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_mono
tff(fact_2183_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_2184_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_2185_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) ) ).

% diff_strict_right_mono
tff(fact_2186_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_strict_right_mono
tff(fact_2187_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_2188_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)) ) ).

% diff_strict_left_mono
tff(fact_2189_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_strict_left_mono
tff(fact_2190_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_2191_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2192_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2193_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_2194_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ) ) ).

% diff_strict_mono
tff(fact_2195_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_2196_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_2197_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_2198_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_2199_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_2200_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_2201_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_2202_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_2203_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_2204_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_2205_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_2206_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_2207_times__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),W)) ).

% times_divide_times_eq
tff(fact_2208_times__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_2209_times__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_2210_divide__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),W)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z)) ).

% divide_divide_times_eq
tff(fact_2211_divide__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).

% divide_divide_times_eq
tff(fact_2212_divide__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),W)),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2213_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) ).

% divide_divide_eq_left'
tff(fact_2214_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_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_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_2233_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_2234_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_2235_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_2236_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_2237_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_2238_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_2239_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_2240_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_2241_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_2242_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_2243_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_2244_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_2245_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) ).

% add_divide_distrib
tff(fact_2246_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_2247_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_2248_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)),C) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) ).

% diff_divide_distrib
tff(fact_2249_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_2250_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_2251_max__add__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_2252_max__add__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xb),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ).

% max_add_distrib_left
tff(fact_2253_max__add__distrib__left,axiom,
    ! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_2254_max__add__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_2255_max__add__distrib__left,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(ord_max_Code_integer(Xb,Y),Z) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xb,Z),plus_p5714425477246183910nteger(Y,Z)) ).

% max_add_distrib_left
tff(fact_2256_max__add__distrib__right,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2257_max__add__distrib__right,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_rat_rat(plus_plus_rat(Xb),Z)) ).

% max_add_distrib_right
tff(fact_2258_max__add__distrib__right,axiom,
    ! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2259_max__add__distrib__right,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).

% max_add_distrib_right
tff(fact_2260_max__add__distrib__right,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] : plus_p5714425477246183910nteger(Xb,ord_max_Code_integer(Y,Z)) = ord_max_Code_integer(plus_p5714425477246183910nteger(Xb,Y),plus_p5714425477246183910nteger(Xb,Z)) ).

% max_add_distrib_right
tff(fact_2261_max__diff__distrib__left,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] : minus_8373710615458151222nteger(ord_max_Code_integer(Xb,Y),Z) = ord_max_Code_integer(minus_8373710615458151222nteger(Xb,Z),minus_8373710615458151222nteger(Y,Z)) ).

% max_diff_distrib_left
tff(fact_2262_max__diff__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_2263_max__diff__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xb),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xb),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_2264_max__diff__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_2265_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2266_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2267_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2268_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2269_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I2),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2270_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2271_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2272_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2273_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2274_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_2275_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2276_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_2277_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2278_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_2279_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2280_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_2281_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_le_eq
tff(fact_2282_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_le_eq
tff(fact_2283_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_2284_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% le_diff_eq
tff(fact_2285_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% le_diff_eq
tff(fact_2286_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_2287_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_2288_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_2289_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_2290_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_2291_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_2292_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_2293_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_2294_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_2295_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_2296_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_2297_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) ) ).

% diff_less_eq
tff(fact_2298_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_less_eq
tff(fact_2299_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_2300_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ) ).

% less_diff_eq
tff(fact_2301_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% less_diff_eq
tff(fact_2302_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_2303_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_2304_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_2305_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_2306_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_2307_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_2308_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S2)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_2309_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_2310_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2),aa_nat7603221163087113075at_nat(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))),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_2311_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_2312_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R2,aa_num_Code_integer(numera6620942414471956472nteger,L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_2313_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_2314_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw2: $o] : Xb = vEBT_Leaf((A3),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != $ite(
                        (B2),
                        some_nat(one_one_nat),
                        $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              some_nat(Ma),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_2315_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B2),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
                 => ( Y != none_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            some_nat(Mi),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_2316_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_2317_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_2318_vebt__delete_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B2)) ) ) )
       => ( ! [A3: $o] :
              ( ? [B2: $o] : Xb = vEBT_Leaf((A3),(B2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A3),$false) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Y != vEBT_Leaf((A3),(B2)) ) ) )
           => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
                 => ( Y != vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) ) )
             => ( ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry) )
                   => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry) ) )
               => ( ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                     => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma ) ),
                                vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),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)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi,xn,Mi),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary,h),
                                                vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                      $ite(
                                                        xn = Ma,
                                                        $let(
                                                          maxs: option_nat,
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                        Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,sn) ),
                                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                    $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,Summary) ) ) ),
                                        vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_2319_low__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_2320_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_2321_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_2322_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_2323_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_2324_Leaf__0__not,axiom,
    ! [A: $o,B: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_2325_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A3: $o,B2: $o] : Ta = vEBT_Leaf((A3),(B2)) ) ) ).

% deg_1_Leafy
tff(fact_2326_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A3: $o,B2: $o] : Ta = vEBT_Leaf((A3),(B2)) ) ).

% deg_1_Leaf
tff(fact_2327_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ).

% deg1Leaf
tff(fact_2328_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xb)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),Xb) ) ) ).

% both_member_options_def
tff(fact_2329_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_2330_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_2331_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_2332_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mod_trivial
tff(fact_2333_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_2334_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xb)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_2335_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y222)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y222) ) ) ) ).

% VEBT.inject(2)
tff(fact_2336_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2337_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2338_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_2339_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_2340_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_2341_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_2342_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_2343_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_2344_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_2345_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_2346_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_2347_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_2348_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_2349_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_2350_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_2351_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_2352_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_2353_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_2354_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_2355_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2356_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_2357_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_2358_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_2359_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_2360_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_2361_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_2362_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_2363_add__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A) = A ).

% add_0
tff(fact_2364_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_2365_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_2366_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_2367_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_2368_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2369_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2370_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_2371_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_2372_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_2373_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_2374_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_2375_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_2376_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_2377_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_2378_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_2379_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_2380_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_2381_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_2382_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_2383_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_2384_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_2385_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_2394_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_2395_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_2396_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_2397_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_2398_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),A) = zero_zero_complex ).

% diff_self
tff(fact_2399_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_2400_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_2401_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_2402_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_2403_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_2404_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_2405_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_2406_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_2407_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),zero_zero_complex) = A ).

% diff_zero
tff(fact_2408_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_2409_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_2410_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_2411_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_2412_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2413_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_2414_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_2415_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_2416_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_2417_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_2418_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2419_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_2420_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_2421_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_2422_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_2423_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_2424_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_2425_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_2426_div__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_2427_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_2428_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_2429_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_2430_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_2431_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_2432_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_2433_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_2434_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_2435_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_2436_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_2437_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,C),A) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_2438_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_2439_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_2440_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_2441_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_2442_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_2443_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_2444_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_2445_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_2446_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_2447_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_2448_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_2449_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_2450_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_2451_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_2452_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_2453_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_2454_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_2455_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_2456_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_2457_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_2458_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).

% Nat.add_0_right
tff(fact_2459_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_2460_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_2461_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_2462_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_2463_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_2464_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_2465_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_2466_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_2467_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_2468_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2469_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2470_mult__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2471_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2472_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mod_self2
tff(fact_2473_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_2474_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_2475_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_2476_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = Mb ) ) ).

% mod_less
tff(fact_2477_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_2478_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_2479_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_2480_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_2481_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_2482_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_2483_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2484_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2485_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2486_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2487_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2488_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2489_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_2490_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_2491_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_2492_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_2493_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_2494_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_2495_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_2496_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_2497_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_2498_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2499_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2500_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2501_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_2502_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2503_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2504_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2505_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2506_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2507_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2508_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2509_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2510_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2511_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_2512_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_2513_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_2514_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_2515_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_2516_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_2517_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_2518_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_2519_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_2520_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_2521_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_2522_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_2523_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_2524_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_2525_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_2526_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_2527_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2528_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2529_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_2530_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2531_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2532_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_2533_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2534_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2535_sum__squares__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2536_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2537_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_2538_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_2539_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_2540_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2541_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_2542_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_2543_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_2544_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2545_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_2546_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_2547_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_2548_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2549_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_2550_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_2551_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_2552_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2553_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_2554_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_2555_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_2556_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2557_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2558_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_2559_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_2560_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_2561_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2562_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_2563_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_2564_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_2565_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_2566_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_2567_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_2568_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2569_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_2570_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_2571_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2572_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_2573_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_2574_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2575_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_2576_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_2577_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2578_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_2579_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_2580_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2581_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2582_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2583_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_2584_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_2585_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_2586_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_2587_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_2588_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),A) = one_one_complex ) ) ).

% div_self
tff(fact_2589_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_2590_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2591_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_2592_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_2593_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_2594_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_2595_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_2596_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_2597_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_2598_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_2599_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_2600_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_2601_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_2602_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2603_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(divide_divide_rat(A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2604_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_2605_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_2606_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2607_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2608_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_2609_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_2610_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2611_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_2612_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_2613_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_2614_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_2615_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_2616_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_2617_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_2618_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_2619_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_2620_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_2621_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_2622_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_2623_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_2624_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_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_2625_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_2626_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_2627_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_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_2628_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_2629_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_2630_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_2631_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_2632_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_2633_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_2634_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_2635_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_2636_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2637_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2638_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_2639_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_2640_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_2641_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_2642_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_2643_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_2644_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_2645_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_2646_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_2647_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_2648_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_2649_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_2650_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_2651_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_2652_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_2653_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_2654_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_2655_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_2656_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_2657_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_2658_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_2659_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(Xb)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(4)
tff(fact_2660_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Xb),zero_z3403309356797280102nteger) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(4)
tff(fact_2661_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xb)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(4)
tff(fact_2662_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(4)
tff(fact_2663_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(4)
tff(fact_2664_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(4)
tff(fact_2665_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(3)
tff(fact_2666_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb)) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(3)
tff(fact_2667_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(3)
tff(fact_2668_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(3)
tff(fact_2669_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(3)
tff(fact_2670_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(3)
tff(fact_2671_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2672_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2673_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_2674_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_2675_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_2676_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_2677_max__0__1_I1_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_2678_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_2679_max__0__1_I1_J,axiom,
    ord_max_Code_integer(zero_z3403309356797280102nteger,one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_2680_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_2681_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_2682_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_2683_max__0__1_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_2684_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_2685_max__0__1_I2_J,axiom,
    ord_max_Code_integer(one_one_Code_integer,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_2686_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2687_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_2688_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2689_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_2690_mult__less__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_2691_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).

% div_by_Suc_0
tff(fact_2692_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_2693_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_2694_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_2695_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_2696_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_2697_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_2698_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_2699_mod__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_2700_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2701_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2702_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_2703_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_2704_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2705_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2706_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2707_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2708_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_num_real(numeral_numeral_real,W)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2709_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2710_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_2711_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2712_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2713_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2714_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2715_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2716_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2717_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2718_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2719_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2720_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_2721_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_2722_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_2723_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_2724_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_2725_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_2726_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_2727_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_2728_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_2729_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_2730_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2731_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_2732_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_2733_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2734_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_2735_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_2736_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2737_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2738_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2739_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2740_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_2741_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_2742_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_2743_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2744_mult__le__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_2745_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_2746_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_2747_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_2748_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_2749_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_2750_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_2751_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2752_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2753_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2754_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2755_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2756_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2757_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2758_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2759_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_2760_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_2761_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_2762_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(aa_int_fun_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_2763_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(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_2764_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2765_power__strict__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2766_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2767_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2768_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2769_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2770_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2771_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_2772_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_2773_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_2774_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_2775_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_2776_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_2777_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_2778_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_2779_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_2780_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Nb))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2781_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_2782_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2783_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_2784_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2785_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2786_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(Xb),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))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2787_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,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))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2788_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2789_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_2790_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_2791_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_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_2792_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_2793_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_2794_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_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_2795_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2796_power__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2797_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2798_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_2799_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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 )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2800_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2801_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2802_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_2803_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2804_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2805_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_2806_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2807_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2808_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2809_add__self__mod__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_2810_mod2__gr__0,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_2811_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2812_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2813_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2814_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2815_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2816_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2817_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_2818_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_2819_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_2820_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q3: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_2821_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2822_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_2823_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2824_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2825_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D3)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_2826_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2827_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_2828_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),zero_zero_nat) = vEBT_Leaf($false,(B)) ).

% vebt_delete.simps(1)
tff(fact_2829_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),X2)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2830_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_2831_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2832_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2833_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2834_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2835_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2836_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2837_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),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,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2838_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2839_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2840_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_2841_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_2842_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_2843_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_2844_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : Xb != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2845_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : Xb != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : Xb != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B2: nat] : Xb != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A3),some_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2846_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : Xb != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : Xb != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A3: num,B2: num] : Xb != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A3),some_num(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_2847_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv2: option4927543243414619207at_nat] : Xb != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2028404973800837934_nat_o,X2: product_prod_nat_nat,Y4: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X2),some_P7363390416028606310at_nat(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2848_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv2: option_nat] : Xb != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_o,V2: nat] : Xb != produc7967433646104525454on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X2: nat,Y4: nat] : Xb != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X2),some_nat(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2849_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc50614890452615238on_num] :
      ( ! [Uu2: fun_num_fun_num_o,Uv2: option_num] : Xb != produc1140783415258738358on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_o,V2: num] : Xb != produc1140783415258738358on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X2: num,Y4: num] : Xb != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X2),some_num(Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_2850_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_2851_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((A),$false) ).

% vebt_delete.simps(2)
tff(fact_2852_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_2853_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_2854_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_2855_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_2856_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_2857_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_2858_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_2859_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_2860_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_2861_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_2862_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_2863_mod__mult__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(times_times_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(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_2864_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_right_eq
tff(fact_2865_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_right_eq
tff(fact_2866_mod__mult__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(times_times_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(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_2867_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_left_eq
tff(fact_2868_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_left_eq
tff(fact_2869_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_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_mod_right
tff(fact_2870_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_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_mod_right
tff(fact_2871_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_2872_mod__mult__mult2,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(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(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_2873_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),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,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).

% mod_mult_mult2
tff(fact_2874_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_2875_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B5: 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,A5),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,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_2876_mod__mult__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A5),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_2877_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A5,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A5),B5),C) ) ) ) ).

% mod_mult_cong
tff(fact_2878_mod__mult__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(times_times_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(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_2879_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_2880_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_eq
tff(fact_2881_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_2882_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_2883_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_2884_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_2885_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_2886_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_2887_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B5: 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,A5),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,B5),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,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_2888_mod__add__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_2889_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A5,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A5,B5),C) ) ) ) ).

% mod_add_cong
tff(fact_2890_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_2891_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_2892_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_2893_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_right_eq
tff(fact_2894_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_2895_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_left_eq
tff(fact_2896_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_2897_mod__diff__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A5),B5)),C) ) ) ) ).

% mod_diff_cong
tff(fact_2898_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B5: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A5,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B5,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A5,B5),C) ) ) ) ).

% mod_diff_cong
tff(fact_2899_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_2900_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_2901_power__mod,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_2902_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),Nb)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),B) ).

% power_mod
tff(fact_2903_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),Nb),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),B) ).

% power_mod
tff(fact_2904_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_2905_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_2906_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_2907_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_2908_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_2909_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_2910_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_2911_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_2912_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_2913_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_2914_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_2915_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_2916_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_2917_zero__le,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).

% zero_le
tff(fact_2918_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_2919_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_2920_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_2921_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_2922_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2923_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [E2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2924_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_2925_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_2926_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_2927_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_2928_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_2929_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_2930_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_2931_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_2932_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_2933_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2934_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2935_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2936_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_2937_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_2938_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,one)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% cong_exp_iff_simps(4)
tff(fact_2939_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_2940_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_2941_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_2942_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_2943_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_2944_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_2945_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_2946_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_2947_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_2948_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_2949_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_2950_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_2951_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_2952_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_2953_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_2954_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_2955_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_2956_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_2957_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_2958_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_2959_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_2960_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_2961_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_2962_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_2963_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_2964_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_2965_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_2966_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_2967_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_2968_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_2969_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_2970_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_2971_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_2972_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_2973_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2974_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_2975_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_2976_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_2977_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_2978_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2979_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2980_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2981_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2982_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2983_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_2984_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_2985_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_2986_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_2987_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_2988_power__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_2989_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_2990_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_2991_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_2992_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_2993_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_2994_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_2995_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_2996_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_2997_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_2998_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_2999_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_3000_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_3001_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_3002_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_3003_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_3004_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X2),zero_zero_nat)
     => ( ! [Y4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y4))
       => ( ! [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X2),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_3005_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_3006_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_3007_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_3008_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_3009_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% not0_implies_Suc
tff(fact_3010_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_3011_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_3012_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_3013_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_3014_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_3015_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_3016_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_3017_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_3018_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_3019_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_3020_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_3021_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_3022_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_3023_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_3024_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_3025_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).

% minus_nat.diff_0
tff(fact_3026_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_3027_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_3028_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_3029_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3030_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3031_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3032_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I5)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_3033_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3034_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3035_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3036_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3037_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3038_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3039_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3040_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3041_lambda__zero,axiom,
    aTP_Lamm_bi = aa_com3342990543769683525omplex(times_times_complex,zero_zero_complex) ).

% lambda_zero
tff(fact_3042_lambda__zero,axiom,
    aTP_Lamm_bj = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_3043_lambda__zero,axiom,
    aTP_Lamm_bk = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_3044_lambda__zero,axiom,
    aTP_Lamm_bl = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3045_lambda__zero,axiom,
    aTP_Lamm_bm = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3046_vebt__insert_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),X2)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S),X2)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S),X2)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),X2)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X2) ) ) ) ) ).

% vebt_insert.cases
tff(fact_3047_vebt__delete_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),zero_zero_nat)
     => ( ! [A3: $o,B2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B2: $o,N: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,N)))
         => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary),Uu2)
           => ( ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry),X2)
             => ( ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),X2)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X2) ) ) ) ) ) ) ).

% vebt_delete.cases
tff(fact_3048_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),X2)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),X2)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3049_vebt__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),X2)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X2)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X2)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X2)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X2) ) ) ) ) ).

% vebt_member.cases
tff(fact_3050_vebt__succ_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2)),zero_zero_nat)
     => ( ! [Uv2: $o,Uw2: $o,N: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve)
           => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X2) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_3051_vebt__pred_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)
     => ( ! [A3: $o,Uw2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B2: $o,Va2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Vf)
             => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),X2) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_3052_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,Va,Vb)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3053_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3054_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3055_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3056_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,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(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_3057_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_3058_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_3059_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),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))) ).

% cong_exp_iff_simps(8)
tff(fact_3060_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_3061_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_3062_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D3: 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),D3)) ) ).

% mod_eqE
tff(fact_3063_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D3: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D3)) ) ).

% mod_eqE
tff(fact_3064_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_3065_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_3066_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_3067_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3068_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
           => aa_nat_o(P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_3069_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_3070_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_3071_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat2E
tff(fact_3072_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ~ ! [S: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat1E
tff(fact_3073_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3074_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3075_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3076_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_3077_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Nb: nat] : vEBT_vebt_delete(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = vEBT_Leaf((A),(B)) ).

% vebt_delete.simps(3)
tff(fact_3078_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ).

% mod_if
tff(fact_3079_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_3080_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A),(B))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_3081_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_3082_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_3083_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3084_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3085_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A),(B))) = $ite(
        (A),
        some_nat(zero_zero_nat),
        $ite((B),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_3086_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3087_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3088_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A),(Uw)),aa_nat_nat(suc,zero_zero_nat)) = $ite((A),some_nat(zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_3089_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B)),zero_zero_nat) = $ite((B),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_3090_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_3091_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_3092_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_3093_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_3094_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_3095_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_3096_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_3097_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_3098_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3099_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3100_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3101_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_3102_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_3103_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_3104_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_3105_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3106_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3107_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3108_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_3109_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_3110_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3111_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_3112_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_3113_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_3114_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3115_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_3116_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_3117_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3118_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3119_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3120_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_3121_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_3122_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_3123_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_3124_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_3125_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3126_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3127_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_3128_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono
tff(fact_3129_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono
tff(fact_3130_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_3131_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_3132_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3133_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3134_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_3135_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono
tff(fact_3136_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono
tff(fact_3137_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_3138_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_3139_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3140_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3141_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_3142_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3143_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3144_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_3145_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ).

% split_mult_pos_le
tff(fact_3146_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_3147_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_3148_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)) ).

% zero_le_square
tff(fact_3149_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_3150_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_3151_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3152_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3153_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3154_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_3155_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3156_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3157_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3158_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_3159_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_3160_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_3161_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_3162_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_3163_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_3164_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_3165_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_3166_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_3167_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3168_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3169_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3170_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3171_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3172_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3173_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3174_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_3175_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_3176_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_3177_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_3178_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_3179_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3180_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3181_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3182_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_3183_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_3184_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing2
tff(fact_3185_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_3186_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_3187_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3188_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3189_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3190_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_3191_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_3192_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing
tff(fact_3193_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_3194_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_3195_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3196_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing
tff(fact_3197_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3198_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_3199_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3200_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3201_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3202_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_3203_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3204_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3205_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_3206_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3207_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3208_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3209_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_3210_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3211_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3212_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_3213_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3214_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3215_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_3216_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3217_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3218_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3219_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_3220_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3221_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3222_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_3223_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3224_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3225_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_3226_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3227_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3228_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_3229_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3230_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3231_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3232_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_3233_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_3234_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3235_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_3236_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_3237_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_3238_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_3239_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_3240_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_3241_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_3242_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_3243_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_3244_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3245_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_3246_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3247_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_3248_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_3249_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_3250_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_3251_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_3252_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_3253_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_3254_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_3255_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_3256_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3257_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3258_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_3259_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_3260_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_3261_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_3262_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3263_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_3264_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_3265_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_3266_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_3267_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_3268_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_3269_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3270_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3271_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3272_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3273_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_3274_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_3275_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_3276_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_3277_add__less__zeroD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_3278_add__less__zeroD,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xb),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_3279_add__less__zeroD,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_3280_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_3281_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% pos_add_strict
tff(fact_3282_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_3283_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_3284_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_3285_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_pos
tff(fact_3286_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_3287_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_3288_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_3289_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_3290_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_3291_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_3292_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_3293_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_3294_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_3295_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_3296_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_3297_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_3298_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_3299_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_3300_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_3301_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_3302_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_3303_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_3304_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_3305_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_3306_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3307_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_3308_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3309_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3310_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3311_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3312_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_3313_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3314_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_3315_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3316_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3317_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3318_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_3319_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3320_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_right_mono
tff(fact_3321_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_right_mono
tff(fact_3322_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3323_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_3324_zero__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_power
tff(fact_3325_zero__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_power
tff(fact_3326_zero__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_le_power
tff(fact_3327_zero__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_le_power
tff(fact_3328_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_3329_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_3330_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_3331_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono
tff(fact_3332_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3333_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3334_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_3335_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3336_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3337_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_3338_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_3339_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_3340_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_3341_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_3342_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_3343_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_3344_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_3345_divide__pos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_3346_divide__pos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_3347_divide__pos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_3348_divide__pos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_3349_divide__neg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_3350_divide__neg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_3351_divide__neg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_3352_divide__neg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_3353_zero__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_less_power
tff(fact_3354_zero__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_less_power
tff(fact_3355_zero__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_less_power
tff(fact_3356_zero__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_less_power
tff(fact_3357_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
      <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_3358_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_3359_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_3360_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = A )
      <=> ( B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_3361_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_3362_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_3363_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = B )
       => ( A = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_3364_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_3365_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_3366_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_3367_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_3368_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_3369_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_3370_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_3371_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_3372_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_3373_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_3374_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_3375_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z) )
        <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3376_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3377_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xb),Y) = aa_rat_rat(divide_divide_rat(W),Z) )
        <=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_3378_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_3379_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_3380_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_3381_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3382_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_3383_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3384_power__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3385_power__0,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3386_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3387_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3388_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3389_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I5)) ) ) ) ).

% Ex_less_Suc2
tff(fact_3390_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M5: nat] : Nb = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_3391_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I5)) ) ) ) ).

% All_less_Suc2
tff(fact_3392_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M: nat] : Nb = aa_nat_nat(suc,M) ) ).

% gr0_implies_Suc
tff(fact_3393_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_3394_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_3395_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_3396_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_3397_option_Osize_I4_J,axiom,
    ! [X22: nat] : size_size_option_nat(some_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3398_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_3399_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_3400_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
               => ~ aa_nat_o(P,I) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_3401_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3402_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3403_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3404_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_3405_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3406_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_3407_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_3408_nat__power__less__imp__less,axiom,
    ! [I2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_3409_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_3410_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3411_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3412_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_3413_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_3414_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_3415_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_3416_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3417_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts2,S2),Xb) = vEBT_Node(Info,zero_zero_nat,Ts2,S2) ).

% vebt_insert.simps(2)
tff(fact_3418_vebt__pred_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A),(B)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        (B),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_3419_mod__double__modulus,axiom,
    ! [Mb: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
       => ( ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = modulo364778990260209775nteger(Xb,Mb) )
          | ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xb,Mb),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3420_mod__double__modulus,axiom,
    ! [Mb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3421_mod__double__modulus,axiom,
    ! [Mb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3422_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3423_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3424_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3425_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_3426_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_3427_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_3428_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_3429_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_3430_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_3431_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_3432_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3433_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_3434_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_3435_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_3436_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_3437_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_3438_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_3439_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_3440_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_3441_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_3442_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_3443_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_3444_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_3445_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_3446_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_3447_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_3448_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_3449_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_3450_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(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mult_div_eq_mod
tff(fact_3451_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_3452_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_3453_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_3454_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_3455_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_3456_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_3457_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_3458_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_3459_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(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_div_mult_eq_mod
tff(fact_3460_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_3461_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_3462_modulo__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_3463_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3464_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_3465_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3466_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_3467_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3468_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3469_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3470_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_3471_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3472_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3473_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_3474_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3475_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3476_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3477_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_3478_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3479_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3480_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3481_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_3482_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3483_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3484_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_3485_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3486_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3487_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3488_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_3489_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3490_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3491_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3492_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_3493_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3494_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3495_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_3496_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3497_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3498_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_3499_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3500_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3501_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_3502_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3503_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3504_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3505_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_3506_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3507_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3508_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3509_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_3510_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3511_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3512_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3513_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_3514_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3515_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3516_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3517_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_3518_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3519_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3520_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3521_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_3522_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3523_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing
tff(fact_3524_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3525_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_3526_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3527_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3528_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3529_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_3530_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_3531_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_3532_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_3533_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_3534_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3535_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3536_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3537_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_3538_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_3539_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_3540_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_3541_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_3542_field__le__epsilon,axiom,
    ! [Xb: real,Y: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_epsilon
tff(fact_3543_field__le__epsilon,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_epsilon
tff(fact_3544_sum__squares__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3545_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3546_sum__squares__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_3547_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3548_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3549_sum__squares__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_3550_divide__nonpos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_3551_divide__nonpos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3552_divide__nonpos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3553_divide__nonpos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3554_divide__nonneg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3555_divide__nonneg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3556_divide__nonneg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_3557_divide__nonneg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3558_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3559_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_3560_frac__less2,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_3561_frac__less2,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_3562_frac__less,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_3563_frac__less,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_less
tff(fact_3564_frac__le,axiom,
    ! [Y: real,Xb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_3565_frac__le,axiom,
    ! [Y: rat,Xb: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_le
tff(fact_3566_mult__left__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3567_mult__left__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3568_mult__left__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_3569_mult__right__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3570_mult__right__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3571_mult__right__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_3572_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3573_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3574_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3575_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3576_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3577_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_3578_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3579_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_3580_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3581_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3582_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3583_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_3584_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_3585_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_3586_not__sum__squares__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_3587_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3588_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3589_sum__squares__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_3590_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_3591_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_3592_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_3593_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_3594_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3595_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_3596_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3597_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_3598_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3599_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_3600_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3601_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_3602_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3603_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_3604_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3605_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_3606_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3607_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_3608_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3609_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_3610_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_3611_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_3612_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_3613_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_3614_power__le__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_3615_power__le__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_3616_power__le__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3617_power__le__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_3618_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3619_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C) = B,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3620_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3621_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3622_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3623_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3624_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3625_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3626_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3627_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_3628_divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z)),Y) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3629_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: 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,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3630_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_3631_add__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Y),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3632_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),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,Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3633_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xb),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(Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_3634_add__num__frac,axiom,
    ! [Y: complex,Z: complex,Xb: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3635_add__num__frac,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3636_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xb),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_3637_add__frac__num,axiom,
    ! [Y: complex,Xb: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)),Z) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3638_add__frac__num,axiom,
    ! [Y: real,Xb: 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,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3639_add__frac__num,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_3640_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3641_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: 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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),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,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3642_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_3643_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3644_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3645_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_3646_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),Z)),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3647_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3648_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_3649_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3650_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3651_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3652_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_3653_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3654_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3655_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3656_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_3657_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_3658_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3659_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_3660_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3661_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z)),Y) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3662_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: 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,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3663_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3664_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Y),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3665_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),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,Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3666_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xb),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(Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3667_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,W),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Z)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,W),Y))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3668_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: 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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),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,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3669_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3670_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),Z)) = $ite(Z = zero_zero_complex,A,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3671_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3672_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3673_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_3674_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_3675_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
               => ~ aa_nat_o(P,I) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_3676_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_3677_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_3678_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_3679_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K)) ) ).

% power_gt_expt
tff(fact_3680_diff__Suc__less,axiom,
    ! [Nb: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I2))),Nb) ) ).

% diff_Suc_less
tff(fact_3681_nat__one__le__power,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Nb)) ) ).

% nat_one_le_power
tff(fact_3682_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_3683_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_3684_length__pos__if__in__set,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3685_length__pos__if__in__set,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3686_length__pos__if__in__set,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xb,set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3687_length__pos__if__in__set,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3688_length__pos__if__in__set,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3689_length__pos__if__in__set,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3690_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( member_int(Xb,set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3691_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split
tff(fact_3692_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_3693_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_3694_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Mb)) ) ) ).

% div_le_mono2
tff(fact_3695_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_3696_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_3697_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_3698_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Mb )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3699_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3700_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3701_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3702_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S2) ).

% vebt_insert.simps(3)
tff(fact_3703_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_3704_vebt__mint_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A3: $o,B2: $o] : Xb != vEBT_Leaf((A3),(B2))
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_3705_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3706_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( Y != $ite(
                  (B2),
                  some_nat(one_one_nat),
                  $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3707_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( Y != $ite(
                  (A3),
                  some_nat(zero_zero_nat),
                  $ite((B2),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3708_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3709_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3710_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3711_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3712_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3713_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3714_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3715_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3716_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3717_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3718_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3719_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3720_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3721_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3722_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3723_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3724_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3725_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3726_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3727_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3728_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3729_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3730_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3731_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3732_field__le__mult__one__interval,axiom,
    ! [Xb: real,Y: real] :
      ( ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),Xb)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_3733_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z3),Xb)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_3734_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3735_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_3736_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3737_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_3738_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3739_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_3740_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3741_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_3742_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3743_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_3744_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3745_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_3746_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3747_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_3748_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3749_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_3750_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_3751_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_3752_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_3753_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_3754_convex__bound__le,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3755_convex__bound__le,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3756_convex__bound__le,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3757_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3758_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3759_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3760_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_3761_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3762_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3763_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3764_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3765_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_3766_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3767_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3768_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3769_power__Suc__less,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3770_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_3771_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_3772_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3773_power__Suc__le__self,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3774_power__Suc__le__self,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3775_power__Suc__le__self,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3776_power__Suc__le__self,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_3777_power__Suc__less__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_3778_power__Suc__less__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_3779_power__Suc__less__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_3780_power__Suc__less__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_3781_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_3782_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_3783_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_3784_zero__power2,axiom,
    aa_nat_int(aa_int_fun_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_3785_zero__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_3786_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3787_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3788_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3789_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3790_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3791_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3792_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3793_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3794_self__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% self_le_power
tff(fact_3795_self__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_3796_self__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% self_le_power
tff(fact_3797_self__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% self_le_power
tff(fact_3798_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_3799_one__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% one_less_power
tff(fact_3800_one__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_3801_one__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_3802_one__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% one_less_power
tff(fact_3803_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_3804_power__diff,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3805_power__diff,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3806_power__diff,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3807_power__diff,axiom,
    ! [A: nat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3808_power__diff,axiom,
    ! [A: int,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3809_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_3810_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_3811_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_3812_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I5)),J3) )
               => aa_nat_o(P,I5) ) ) ) ) ) ).

% split_div
tff(fact_3813_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3814_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3815_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3816_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3817_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3818_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3819_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_3820_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_3821_vebt__delete_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TrLst2,Smry2),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TrLst2,Smry2) ).

% vebt_delete.simps(5)
tff(fact_3822_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_3823_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve2) = none_nat ).

% vebt_succ.simps(4)
tff(fact_3824_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve2),Vf2) = none_nat ).

% vebt_pred.simps(5)
tff(fact_3825_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3826_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3827_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3828_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3829_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3830_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3831_convex__bound__lt,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3832_convex__bound__lt,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3833_convex__bound__lt,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3834_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_3835_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_3836_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_3837_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_3838_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_3839_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_3840_power2__le__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3841_power2__le__imp__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3842_power2__le__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3843_power2__le__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3844_power2__eq__imp__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3845_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3846_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3847_power2__eq__imp__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3848_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3849_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3850_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3851_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_3852_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_3853_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_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_3854_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3855_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3856_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3857_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3858_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3859_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3860_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3861_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3862_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_3863_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_3864_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_3865_power__eq__if,axiom,
    ! [P2: complex,Mb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,P2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3866_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3867_power__eq__if,axiom,
    ! [P2: rat,Mb: nat] :
      aa_nat_rat(power_power_rat(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3868_power__eq__if,axiom,
    ! [P2: nat,Mb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3869_power__eq__if,axiom,
    ! [P2: int,Mb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3870_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) ) ) ).

% power_minus_mult
tff(fact_3871_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_minus_mult
tff(fact_3872_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3873_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_3874_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) ) ) ).

% power_minus_mult
tff(fact_3875_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_3876_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_3877_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_3878_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Mb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_3879_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,Mb: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_3880_vebt__delete_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr2,Sm2) ).

% vebt_delete.simps(6)
tff(fact_3881_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2) = none_nat ).

% vebt_succ.simps(5)
tff(fact_3882_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2) = none_nat ).

% vebt_pred.simps(6)
tff(fact_3883_power2__less__imp__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3884_power2__less__imp__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3885_power2__less__imp__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3886_power2__less__imp__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3887_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3888_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3889_sum__power2__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3890_sum__power2__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3891_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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_3892_sum__power2__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3893_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3894_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3895_sum__power2__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3896_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_3897_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),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_3898_not__sum__power2__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_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_3899_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3900_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3901_zero__le__even__power_H,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_3902_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_3903_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_3904_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_3905_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3906_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3907_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_3908_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3909_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3910_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3911_odd__power__less__zero,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_3912_odd__power__less__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_3913_odd__power__less__zero,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_3914_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3915_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_3916_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3917_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3918_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3919_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3920_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3921_arith__geo__mean,axiom,
    ! [U: rat,Xb: 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(Xb),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_3922_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A4: $o,B4: $o] : A1 = vEBT_Leaf((A4),(B4))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( 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))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(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))),N4))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I5) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I5: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(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))),N4))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I5 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I5 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(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,N4)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I5) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I5: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(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,N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I5 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I5 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3923_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A3: $o,B2: $o] : A1 = vEBT_Leaf((A3),(B2))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
                     => ( ( M = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(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) )
                             => ( ! [I: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(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) )
                               => ( ! [I: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3924_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A3),(B2)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B2)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B2))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts,S) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3925_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_3926_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_3927_set__bit__0,axiom,
    ! [A: 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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3928_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_3929_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B2),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B2)),zero_zero_nat)) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                some_nat(Mi),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3930_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A3),some_nat(zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = $ite(
                              (B2),
                              some_nat(one_one_nat),
                              $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  some_nat(Ma),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_3931_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_3932_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_3933_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_3934_vebt__delete_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B2)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A3),(B2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A3),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A3),(B2)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Y = vEBT_Leaf((A3),(B2)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
                   => ( ( Y = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry) )
                     => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst,Smry),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                       => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr,Sm),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),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)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi,xn,Mi),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary,h),
                                                    vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                          $ite(
                                                            xn = Ma,
                                                            $let(
                                                              maxs: option_nat,
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                            Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,sn) ),
                                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                        $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist,Summary) ) ) ),
                                            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_3935_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(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_3936_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_3937_div__mod__decomp,axiom,
    ! [A2: nat,Nb: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb)) ).

% div_mod_decomp
tff(fact_3938_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_3939_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_3940_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_3941_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_3942_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_3943_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3944_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_3945_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_3946_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_3947_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_3948_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_3949_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_3950_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3951_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3952_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3953_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3954_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3955_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3956_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3957_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_3958_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I5: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Nb = 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) ) )
           => aa_int_o(aa_int_fun_int_o(P,I5),J3) ) ) ) ).

% split_pos_lemma
tff(fact_3959_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I5: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Nb = 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) ) )
           => aa_int_o(aa_int_fun_int_o(P,I5),J3) ) ) ) ).

% split_neg_lemma
tff(fact_3960_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_3961_zdiv__mono2__neg,axiom,
    ! [A: int,B5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B5)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3962_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A5),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_3963_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) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3964_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) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3965_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3966_zdiv__mono2,axiom,
    ! [A: int,B5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B5)) ) ) ) ).

% zdiv_mono2
tff(fact_3967_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A5),B)) ) ) ).

% zdiv_mono1
tff(fact_3968_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I5: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Nb = 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) ) )
             => aa_int_o(P,I5) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I5: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = 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) ) )
             => aa_int_o(P,I5) ) ) ) ) ).

% split_zdiv
tff(fact_3969_div__mod__decomp__int,axiom,
    ! [A2: int,Nb: 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(aa_int_fun_int_int(divide_divide_int,A2),Nb)),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Nb)) ).

% div_mod_decomp_int
tff(fact_3970_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3971_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_3972_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,times_7803423173614009249d_enat(Mb,Nb))
    <=> ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Mb)
        & ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_3973_iadd__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_3974_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Nb,zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_3975_i0__lb,axiom,
    ! [Nb: extended_enat] : ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat,Nb) ).

% i0_lb
tff(fact_3976_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less
tff(fact_3977_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less
tff(fact_3978_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_3979_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3980_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3981_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_3982_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_3983_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_3984_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_3985_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ord_less_eq_set_int(A,A) ).

% verit_comp_simplify1(2)
tff(fact_3986_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3987_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3988_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3989_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_3990_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3991_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3992_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3993_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3994_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_3995_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_3996_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_3997_int__power__div__base,axiom,
    ! [Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K),Mb)),K) = aa_nat_int(aa_int_fun_nat_int(power_power_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3998_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),Nb) = A )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb) = A ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3999_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_4000_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B5),A5)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4001_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A5: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B5),A5)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4002_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B5),A5)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4003_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B5),A5)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4004_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),A5)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_4005_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_4006_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_4007_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_4008_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_4009_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_4010_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_4011_max__def__raw,axiom,
    ! [X: extended_enat,Xa: extended_enat] :
      aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Xa) = $ite(ord_le2932123472753598470d_enat(X,Xa),Xa,X) ).

% max_def_raw
tff(fact_4012_max__def__raw,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ord_max_Code_integer(X,Xa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4013_max__def__raw,axiom,
    ! [X: set_int,Xa: set_int] :
      ord_max_set_int(X,Xa) = $ite(ord_less_eq_set_int(X,Xa),Xa,X) ).

% max_def_raw
tff(fact_4014_max__def__raw,axiom,
    ! [X: rat,Xa: rat] :
      aa_rat_rat(ord_max_rat(X),Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4015_max__def__raw,axiom,
    ! [X: num,Xa: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4016_max__def__raw,axiom,
    ! [X: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4017_max__def__raw,axiom,
    ! [X: int,Xa: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_4018_div__less__mono,axiom,
    ! [A2: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_4019_set__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4020_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4021_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_4022_unset__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4023_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4024_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_4025_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B2)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B2))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts,S) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_4026_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4027_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4028_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4029_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4030_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4031_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A3),(B2)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList,Summary),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4032_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4033_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q2),zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_4034_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_4035_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_4036_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_4037_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_4038_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_4039_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_4040_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_4041_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_4042_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_4043_int__less__induct,axiom,
    ! [I2: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_less_induct
tff(fact_4044_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_4045_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4046_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% neg_mod_bound
tff(fact_4047_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_4048_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_4049_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_4050_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I5: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Nb = 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) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I5: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = 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) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_4051_q__pos__lemma,axiom,
    ! [B5: int,Q5: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_4052_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_4053_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_4054_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) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_4055_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) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_4056_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B5: int,Q5: 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,B5),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_4057_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I2),K) = I2 )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2) ) ) ) ).

% zmod_trivial_iff
tff(fact_4058_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B5: int,Q5: 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,B5),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q5)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_4059_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),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))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_4060_zmod__le__nonneg__dividend,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),K)),Mb) ) ).

% zmod_le_nonneg_dividend
tff(fact_4061_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_4062_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_4063_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),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))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_4064_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_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_4065_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_4066_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_4067_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_4068_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_4069_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_4070_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_4071_int__gr__induct,axiom,
    ! [K: int,I2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_gr_induct
tff(fact_4072_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_4073_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_4074_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_4075_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_4076_zmod__eq__0__iff,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
    <=> ? [Q4: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_4077_zmod__eq__0D,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
     => ? [Q3: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_4078_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_4079_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_4080_imult__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_4081_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: 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)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_4082_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),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,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_4083_int__ge__induct,axiom,
    ! [K: int,I2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_ge_induct
tff(fact_4084_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K)) ).

% set_bit_greater_eq
tff(fact_4085_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_4086_verit__la__generic,axiom,
    ! [A: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xb)
      | ( A = Xb )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A) ) ).

% verit_la_generic
tff(fact_4087_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),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,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_4088_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: 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)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_4089_int__le__induct,axiom,
    ! [I2: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_le_induct
tff(fact_4090_int__induct,axiom,
    ! [P: fun_int_o,K: int,I2: int] :
      ( aa_int_o(P,K)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
           => ( aa_int_o(P,I3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_induct
tff(fact_4091_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4092_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4093_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_4094_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4095_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4096_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4097_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4098_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))
    <=> ( ~ ord_less_eq_set_int(A,B)
        | ( ord_less_eq_set_int(C,A)
          & ord_less_eq_set_int(B,D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4099_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4100_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4101_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4102_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ord_less_eq_set_int(set_or1266510415728281911st_int(A,B),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4103_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4104_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( set_or370866239135849197et_int(A,B) = bot_bot_set_set_int )
    <=> ~ ord_less_eq_set_int(A,B) ) ).

% atLeastatMost_empty_iff
tff(fact_4105_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4106_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4107_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( set_or1269000886237332187st_nat(A,B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4108_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4109_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4110_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(A,B) )
    <=> ~ ord_less_eq_set_int(A,B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4111_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4112_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4113_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(A,B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4114_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4115_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4116_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(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_4117_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(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_4118_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 ) )
        | ( ~ ord_less_eq_set_int(L,H)
          & ~ ord_less_eq_set_int(L2,H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4119_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4120_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4121_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = set_or1269000886237332187st_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4122_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4123_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4124_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I2,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I2)
        & aa_set_nat_o(ord_less_eq_set_nat(I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4125_atLeastAtMost__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( member_set_int(I2,set_or370866239135849197et_int(L,U))
    <=> ( ord_less_eq_set_int(L,I2)
        & ord_less_eq_set_int(I2,U) ) ) ).

% atLeastAtMost_iff
tff(fact_4126_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( member_rat(I2,set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4127_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( member_num(I2,set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4128_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I2),set_or1269000886237332187st_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4129_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( member_int(I2,set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4130_atLeastAtMost__iff,axiom,
    ! [I2: real,L: real,U: real] :
      ( member_real(I2,set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4131_aset_I2_J,axiom,
    ! [D2: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D2)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)) ) ) ) ) ) ).

% aset(2)
tff(fact_4132_aset_I1_J,axiom,
    ! [D2: int,A2: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D2)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)) ) ) ) ) ) ).

% aset(1)
tff(fact_4133_bset_I2_J,axiom,
    ! [D2: int,B3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D2)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)) ) ) ) ) ) ).

% bset(2)
tff(fact_4134_bset_I1_J,axiom,
    ! [D2: int,B3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D2)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D2)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)) ) ) ) ) ) ).

% bset(1)
tff(fact_4135_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_4136_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_4137_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_4138_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_4139_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_4140_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_4141_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_4142_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_4143_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_4144_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_4145_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_4146_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_4147_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_4148_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_4149_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_4150_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_4151_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_4152_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_4153_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_4154_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_4155_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% pinf(5)
tff(fact_4156_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% pinf(5)
tff(fact_4157_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% pinf(5)
tff(fact_4158_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% pinf(5)
tff(fact_4159_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% pinf(5)
tff(fact_4160_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% pinf(7)
tff(fact_4161_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% pinf(7)
tff(fact_4162_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% pinf(7)
tff(fact_4163_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% pinf(7)
tff(fact_4164_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% pinf(7)
tff(fact_4165_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_4166_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_4167_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_4168_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_4169_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_4170_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z3: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_4171_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z3: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_4172_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z3: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_4173_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z3: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_4174_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z3: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_4175_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_4176_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_4177_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_4178_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_4179_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_4180_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_4181_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_4182_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_4183_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_4184_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_4185_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% minf(5)
tff(fact_4186_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% minf(5)
tff(fact_4187_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% minf(5)
tff(fact_4188_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% minf(5)
tff(fact_4189_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% minf(5)
tff(fact_4190_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% minf(7)
tff(fact_4191_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% minf(7)
tff(fact_4192_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% minf(7)
tff(fact_4193_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% minf(7)
tff(fact_4194_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% minf(7)
tff(fact_4195_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M6: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X2: nat] :
            ( aa_nat_o(P,X2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M6) )
       => ~ ! [M: nat] :
              ( aa_nat_o(P,M)
             => ~ ! [X: nat] :
                    ( aa_nat_o(P,X)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_4196_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc8114805097830739791at_num] :
      ~ ! [F2: fun_nat_fun_num_num,A3: nat,B2: nat,Acc: num] : Xb != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A3,product_Pair_nat_num(B2,Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4197_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A3: nat,B2: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_4198_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P,X2)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X3: int] :
              ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X3) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4199_bset_I3_J,axiom,
    ! [D2: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int),B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_4200_bset_I4_J,axiom,
    ! [D2: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(Ta,B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_4201_bset_I5_J,axiom,
    ! [D2: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)),Ta) ) ) ) ).

% bset(5)
tff(fact_4202_bset_I7_J,axiom,
    ! [D2: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(Ta,B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)) ) ) ) ) ).

% bset(7)
tff(fact_4203_aset_I3_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int),A2)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_4204_aset_I4_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(Ta,A2)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_4205_aset_I5_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(Ta,A2)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_4206_aset_I7_J,axiom,
    ! [D2: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)) ) ) ) ).

% aset(7)
tff(fact_4207_bset_I6_J,axiom,
    ! [D2: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)),Ta) ) ) ) ).

% bset(6)
tff(fact_4208_bset_I8_J,axiom,
    ! [D2: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int),B3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)) ) ) ) ) ).

% bset(8)
tff(fact_4209_aset_I6_J,axiom,
    ! [D2: int,Ta: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int),A2)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_4210_aset_I8_J,axiom,
    ! [D2: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)) ) ) ) ).

% aset(8)
tff(fact_4211_cppi,axiom,
    ! [D2: int,P: fun_int_o,P3: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D2)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D2))
                    & aa_int_o(P3,X3) )
                | ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D2))
                    & ? [Xa3: int] :
                        ( member_int(Xa3,A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X3)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4212_cpmi,axiom,
    ! [D2: int,P: fun_int_o,P3: fun_int_o,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D2))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B3)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D2)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D2))
                    & aa_int_o(P3,X3) )
                | ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D2))
                    & ? [Xa3: int] :
                        ( member_int(Xa3,B3)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X3)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4213_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% pinf(6)
tff(fact_4214_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% pinf(6)
tff(fact_4215_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% pinf(6)
tff(fact_4216_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% pinf(6)
tff(fact_4217_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% pinf(6)
tff(fact_4218_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% pinf(8)
tff(fact_4219_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% pinf(8)
tff(fact_4220_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% pinf(8)
tff(fact_4221_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% pinf(8)
tff(fact_4222_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% pinf(8)
tff(fact_4223_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% minf(6)
tff(fact_4224_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% minf(6)
tff(fact_4225_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% minf(6)
tff(fact_4226_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% minf(6)
tff(fact_4227_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% minf(6)
tff(fact_4228_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% minf(8)
tff(fact_4229_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% minf(8)
tff(fact_4230_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% minf(8)
tff(fact_4231_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% minf(8)
tff(fact_4232_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% minf(8)
tff(fact_4233_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D2: real,Q: fun_real_o] :
      ( ! [X2: real,K2: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D2))) )
     => ( ! [X2: real,K2: real] :
            ( aa_real_o(Q,X2)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D2))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              | aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D2)))
              | aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D2))) ) ) ) ) ).

% inf_period(2)
tff(fact_4234_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D2: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K2: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K2),D2))) )
     => ( ! [X2: rat,K2: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K2),D2))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              | aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D2)))
              | aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D2))) ) ) ) ) ).

% inf_period(2)
tff(fact_4235_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D2: int,Q: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              | aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2))) ) ) ) ) ).

% inf_period(2)
tff(fact_4236_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D2: real,Q: fun_real_o] :
      ( ! [X2: real,K2: real] :
          ( aa_real_o(P,X2)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D2))) )
     => ( ! [X2: real,K2: real] :
            ( aa_real_o(Q,X2)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D2))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              & aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D2)))
              & aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D2))) ) ) ) ) ).

% inf_period(1)
tff(fact_4237_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D2: rat,Q: fun_rat_o] :
      ( ! [X2: rat,K2: rat] :
          ( aa_rat_o(P,X2)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K2),D2))) )
     => ( ! [X2: rat,K2: rat] :
            ( aa_rat_o(Q,X2)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K2),D2))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              & aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D2)))
              & aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D2))) ) ) ) ) ).

% inf_period(1)
tff(fact_4238_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D2: int,Q: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Q,X2)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              & aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2))) ) ) ) ) ).

% inf_period(1)
tff(fact_4239_conj__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_4240_imp__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_4241_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))
    <=> ( ( ~ ord_less_eq_set_int(A,B)
          | ( ord_less_eq_set_int(C,A)
            & ord_less_eq_set_int(B,D)
            & ( ord_less_set_int(C,A)
              | ord_less_set_int(B,D) ) ) )
        & ord_less_eq_set_int(C,D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4242_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4243_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4244_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(set_or1269000886237332187st_nat(A,B)),set_or1269000886237332187st_nat(C,D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4245_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ord_less_set_int(set_or1266510415728281911st_int(A,B),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4246_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ord_less_set_real(set_or1222579329274155063t_real(A,B),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4247_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P3,X2)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P3,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P3,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_4248_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P1,X2)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P1,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_4249_flip__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4250_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4251_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_4252_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A3: real,B2: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A3),B2)
           => ( aa_real_o(aa_real_fun_real_o(P,B2),C3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A3),C3) ) ) ) )
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [A3: real,B2: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A3)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A3),B2) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_4253_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4254_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xb)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4255_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4256_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4257_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4258_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4259_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_4260_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_4261_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_4262_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_4263_mult__less__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4264_mult__less__iff1,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4265_mult__less__iff1,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4266_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4267_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_4268_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr8326237132889035090at_num(product_nat_num(Xs,Ys),Nb) = product_Pair_nat_num(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_4269_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys),Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys)))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_4270_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr744662078594809490T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys),Nb) = produc599794634098209291T_VEBT(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4271_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr112076138515278198_nat_o(product_nat_o(Xs,Ys),Nb) = product_Pair_nat_o(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_4272_product__nth,axiom,
    ! [Nb: nat,Xs: list_Code_integer,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys),Nb) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_4273_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr3440142176431000676at_int(product_nat_int(Xs,Ys),Nb) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4274_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_4275_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_4276_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_4277_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A2: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,A2)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A2),Xb),X_1) ) ) ) ).

% obtain_set_pred
tff(fact_4278_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A2: set_nat,B3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A2),Z)
       => ( aa_set_nat_o(finite_finite_nat,B3)
         => ( ( A2 = B3 )
           => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A2),Xb),X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_4279_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_4280_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X) ) ) ) ).

% succ_none_empty
tff(fact_4281_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A) ) ) ) ).

% pred_none_empty
tff(fact_4282_prod_Oinject,axiom,
    ! [X1: code_integer,X22: $o,Y1: code_integer,Y22: $o] :
      ( ( produc6677183202524767010eger_o(X1,(X22)) = produc6677183202524767010eger_o(Y1,(Y22)) )
    <=> ( ( X1 = Y1 )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% prod.inject
tff(fact_4283_prod_Oinject,axiom,
    ! [X1: num,X22: num,Y1: num,Y22: num] :
      ( ( product_Pair_num_num(X1,X22) = product_Pair_num_num(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_4284_prod_Oinject,axiom,
    ! [X1: nat,X22: num,Y1: nat,Y22: num] :
      ( ( product_Pair_nat_num(X1,X22) = product_Pair_nat_num(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_4285_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_4286_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_4287_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: $o,A5: code_integer,B5: $o] :
      ( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A5,(B5)) )
    <=> ( ( A = A5 )
        & ( (B)
        <=> (B5) ) ) ) ).

% old.prod.inject
tff(fact_4288_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A5: num,B5: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A5,B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4289_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A5: nat,B5: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A5,B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4290_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A5: nat,B5: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4291_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A5: int,B5: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A5),B5) )
    <=> ( ( A = A5 )
        & ( B = B5 ) ) ) ).

% old.prod.inject
tff(fact_4292_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_4293_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_4294_List_Ofinite__set,axiom,
    ! [Xs: list_int] : finite_finite_int(set_int2(Xs)) ).

% List.finite_set
tff(fact_4295_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : finite3207457112153483333omplex(set_complex2(Xs)) ).

% List.finite_set
tff(fact_4296_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ~ finite_finite_rat(set_or633870826150836451st_rat(A,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% infinite_Icc_iff
tff(fact_4297_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ~ finite_finite_real(set_or1222579329274155063t_real(A,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% infinite_Icc_iff
tff(fact_4298_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_4299_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4300_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_4301_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4302_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4303_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4304_length__product,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT] : size_s6639371672096860321T_VEBT(produc662631939642741121T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_4305_length__product,axiom,
    ! [Xs: list_int,Ys: list_o] : size_s4246224855604898693_int_o(product_int_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_4306_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] : size_s5157815400016825771nt_int(product_int_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4307_finite__nat__set__iff__bounded,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_4308_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Nb) )
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_4309_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_4310_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A2)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).

% finite_list
tff(fact_4311_finite__list,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).

% finite_list
tff(fact_4312_finite__list,axiom,
    ! [A2: set_int] :
      ( finite_finite_int(A2)
     => ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).

% finite_list
tff(fact_4313_finite__list,axiom,
    ! [A2: set_complex] :
      ( finite3207457112153483333omplex(A2)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).

% finite_list
tff(fact_4314_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bn(P),I2))) ).

% finite_M_bounded_by_nat
tff(fact_4315_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bo(F),U))) ) ).

% finite_less_ub
tff(fact_4316_finite__lists__length__eq,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bp(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4317_finite__lists__length__eq,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( finite3207457112153483333omplex(A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bq(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4318_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_br(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4319_finite__lists__length__eq,axiom,
    ! [A2: set_o,Nb: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_bs(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4320_finite__lists__length__eq,axiom,
    ! [A2: set_int,Nb: nat] :
      ( finite_finite_int(A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_bt(A2),Nb))) ) ).

% finite_lists_length_eq
tff(fact_4321_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ finite_finite_rat(set_or633870826150836451st_rat(A,B)) ) ).

% infinite_Icc
tff(fact_4322_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ finite_finite_real(set_or1222579329274155063t_real(A,B)) ) ).

% infinite_Icc
tff(fact_4323_finite__lists__length__le,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bu(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_4324_finite__lists__length__le,axiom,
    ! [A2: set_complex,Nb: nat] :
      ( finite3207457112153483333omplex(A2)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_bv(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_4325_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,Nb: nat] :
      ( finite5795047828879050333T_VEBT(A2)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bw(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_4326_finite__lists__length__le,axiom,
    ! [A2: set_o,Nb: nat] :
      ( finite_finite_o(A2)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_bx(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_4327_finite__lists__length__le,axiom,
    ! [A2: set_int,Nb: nat] :
      ( finite_finite_int(A2)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_by(A2),Nb))) ) ).

% finite_lists_length_le
tff(fact_4328_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_4329_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6271795597528267376eger_o] :
      ~ ! [A3: code_integer,B2: $o] : Y != produc6677183202524767010eger_o(A3,(B2)) ).

% old.prod.exhaust
tff(fact_4330_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_num_num] :
      ~ ! [A3: num,B2: num] : Y != product_Pair_num_num(A3,B2) ).

% old.prod.exhaust
tff(fact_4331_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_num] :
      ~ ! [A3: nat,B2: num] : Y != product_Pair_nat_num(A3,B2) ).

% old.prod.exhaust
tff(fact_4332_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B2: nat] : Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) ).

% old.prod.exhaust
tff(fact_4333_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B2: int] : Y != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B2) ).

% old.prod.exhaust
tff(fact_4334_surj__pair,axiom,
    ! [P2: produc6271795597528267376eger_o] :
    ? [X2: code_integer,Y4: $o] : P2 = produc6677183202524767010eger_o(X2,(Y4)) ).

% surj_pair
tff(fact_4335_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X2: num,Y4: num] : P2 = product_Pair_num_num(X2,Y4) ).

% surj_pair
tff(fact_4336_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X2: nat,Y4: num] : P2 = product_Pair_nat_num(X2,Y4) ).

% surj_pair
tff(fact_4337_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X2: nat,Y4: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4) ).

% surj_pair
tff(fact_4338_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X2: int,Y4: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) ).

% surj_pair
tff(fact_4339_prod__cases,axiom,
    ! [P: fun_Pr3290071492984375169er_o_o,P2: produc6271795597528267376eger_o] :
      ( ! [A3: code_integer,B2: $o] : aa_Pro5286387174248033258er_o_o(P,produc6677183202524767010eger_o(A3,(B2)))
     => aa_Pro5286387174248033258er_o_o(P,P2) ) ).

% prod_cases
tff(fact_4340_prod__cases,axiom,
    ! [P: fun_Pr6331313853612757382_num_o,P2: product_prod_num_num] :
      ( ! [A3: num,B2: num] : aa_Pro7685007914727650815_num_o(P,product_Pair_num_num(A3,B2))
     => aa_Pro7685007914727650815_num_o(P,P2) ) ).

% prod_cases
tff(fact_4341_prod__cases,axiom,
    ! [P: fun_Pr5555124106300582480_num_o,P2: product_prod_nat_num] :
      ( ! [A3: nat,B2: num] : aa_Pro6908818167415475913_num_o(P,product_Pair_nat_num(A3,B2))
     => aa_Pro6908818167415475913_num_o(P,P2) ) ).

% prod_cases
tff(fact_4342_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B2: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_4343_prod__cases,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A3: int,B2: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B2))
     => aa_Pro4592068669816069595_int_o(P,P2) ) ).

% prod_cases
tff(fact_4344_Pair__inject,axiom,
    ! [A: code_integer,B: $o,A5: code_integer,B5: $o] :
      ( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A5,(B5)) )
     => ~ ( ( A = A5 )
         => ( (B)
          <=> ~ (B5) ) ) ) ).

% Pair_inject
tff(fact_4345_Pair__inject,axiom,
    ! [A: num,B: num,A5: num,B5: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A5,B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4346_Pair__inject,axiom,
    ! [A: nat,B: num,A5: nat,B5: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A5,B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4347_Pair__inject,axiom,
    ! [A: nat,B: nat,A5: nat,B5: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A5),B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4348_Pair__inject,axiom,
    ! [A: int,B: int,A5: int,B5: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A5),B5) )
     => ~ ( ( A = A5 )
         => ( B != B5 ) ) ) ).

% Pair_inject
tff(fact_4349_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4350_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4351_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4352_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bz,K))) ).

% finite_Collect_le_nat
tff(fact_4353_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,K))) ).

% finite_Collect_less_nat
tff(fact_4354_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_cb(A2))) ) ).

% finite_Collect_subsets
tff(fact_4355_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( finite3207457112153483333omplex(A2)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_cc(A2))) ) ).

% finite_Collect_subsets
tff(fact_4356_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( finite_finite_int(A2)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_cd(A2))) ) ).

% finite_Collect_subsets
tff(fact_4357_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => finite_finite_real(collect_real(aTP_Lamm_ce(Nb))) ) ).

% finite_roots_unity
tff(fact_4358_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cf,Nb))) ) ).

% finite_roots_unity
tff(fact_4359_finite__Diff2,axiom,
    ! [B3: set_int,A2: set_int] :
      ( finite_finite_int(B3)
     => ( finite_finite_int(minus_minus_set_int(A2,B3))
      <=> finite_finite_int(A2) ) ) ).

% finite_Diff2
tff(fact_4360_finite__Diff2,axiom,
    ! [B3: set_complex,A2: set_complex] :
      ( finite3207457112153483333omplex(B3)
     => ( finite3207457112153483333omplex(minus_811609699411566653omplex(A2,B3))
      <=> finite3207457112153483333omplex(A2) ) ) ).

% finite_Diff2
tff(fact_4361_finite__Diff2,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B3))
      <=> aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_Diff2
tff(fact_4362_finite__Diff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( finite_finite_int(A2)
     => finite_finite_int(minus_minus_set_int(A2,B3)) ) ).

% finite_Diff
tff(fact_4363_finite__Diff,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( finite3207457112153483333omplex(A2)
     => finite3207457112153483333omplex(minus_811609699411566653omplex(A2,B3)) ) ).

% finite_Diff
tff(fact_4364_finite__Diff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A2,B3)) ) ).

% finite_Diff
tff(fact_4365_finite__Collect__disjI,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_cg(P),Q)))
    <=> ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        & finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4366_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ch(P),Q)))
    <=> ( finite1152437895449049373et_nat(collect_set_nat(P))
        & finite1152437895449049373et_nat(collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4367_finite__Collect__disjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ci(P),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4368_finite__Collect__disjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( finite_finite_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cj(P),Q)))
    <=> ( finite_finite_int(collect_int(P))
        & finite_finite_int(collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4369_finite__Collect__disjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ck(P),Q)))
    <=> ( finite3207457112153483333omplex(collect_complex(P))
        & finite3207457112153483333omplex(collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_4370_finite__Collect__conjI,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        | finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) )
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_cl(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4371_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( finite1152437895449049373et_nat(collect_set_nat(P))
        | finite1152437895449049373et_nat(collect_set_nat(Q)) )
     => finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_cm(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4372_finite__Collect__conjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cn(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4373_finite__Collect__conjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ( finite_finite_int(collect_int(P))
        | finite_finite_int(collect_int(Q)) )
     => finite_finite_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_co(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4374_finite__Collect__conjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ( finite3207457112153483333omplex(collect_complex(P))
        | finite3207457112153483333omplex(collect_complex(Q)) )
     => finite3207457112153483333omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_cp(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_4375_finite__interval__int1,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_cq(A),B))) ).

% finite_interval_int1
tff(fact_4376_finite__interval__int4,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_cr(A),B))) ).

% finite_interval_int4
tff(fact_4377_finite__interval__int3,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_cs(A),B))) ).

% finite_interval_int3
tff(fact_4378_finite__interval__int2,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ct(A),B))) ).

% finite_interval_int2
tff(fact_4379_finite__maxlen,axiom,
    ! [M6: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M6)
     => ? [N: nat] :
        ! [X: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X)),N) ) ) ).

% finite_maxlen
tff(fact_4380_finite__maxlen,axiom,
    ! [M6: set_list_o] :
      ( finite_finite_list_o(M6)
     => ? [N: nat] :
        ! [X: list_o] :
          ( member_list_o(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X)),N) ) ) ).

% finite_maxlen
tff(fact_4381_finite__maxlen,axiom,
    ! [M6: set_list_int] :
      ( finite3922522038869484883st_int(M6)
     => ? [N: nat] :
        ! [X: list_int] :
          ( member_list_int(X,M6)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X)),N) ) ) ).

% finite_maxlen
tff(fact_4382_not__finite__existsD,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ~ finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
     => ? [X_1: product_prod_int_int] : aa_Pro4592068669816069595_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_4383_not__finite__existsD,axiom,
    ! [P: fun_set_nat_o] :
      ( ~ finite1152437895449049373et_nat(collect_set_nat(P))
     => ? [X_1: set_nat] : aa_set_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_4384_not__finite__existsD,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ? [X_1: nat] : aa_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_4385_not__finite__existsD,axiom,
    ! [P: fun_int_o] :
      ( ~ finite_finite_int(collect_int(P))
     => ? [X_1: int] : aa_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_4386_not__finite__existsD,axiom,
    ! [P: fun_complex_o] :
      ( ~ finite3207457112153483333omplex(collect_complex(P))
     => ? [X_1: complex] : aa_complex_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_4387_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_nat,R: fun_real_fun_nat_o] :
      ( ~ finite_finite_real(A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B3)
                  & aa_nat_o(aa_real_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B3)
              & ~ finite_finite_real(collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_cu(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4388_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_int,R: fun_real_fun_int_o] :
      ( ~ finite_finite_real(A2)
     => ( finite_finite_int(B3)
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => ? [Xa: int] :
                  ( member_int(Xa,B3)
                  & aa_int_o(aa_real_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( member_int(X2,B3)
              & ~ finite_finite_real(collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_cv(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4389_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B3: set_complex,R: fun_re7319081385389557872plex_o] :
      ( ~ finite_finite_real(A2)
     => ( finite3207457112153483333omplex(B3)
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => ? [Xa: complex] :
                  ( member_complex(Xa,B3)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( member_complex(X2,B3)
              & ~ finite_finite_real(collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_cw(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4390_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_nat,R: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B3)
                  & aa_nat_o(aa_nat_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_cx(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4391_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_int,R: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_finite_int(B3)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
             => ? [Xa: int] :
                  ( member_int(Xa,B3)
                  & aa_int_o(aa_nat_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( member_int(X2,B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_cy(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4392_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B3: set_complex,R: fun_na1554462459066888140plex_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A2)
     => ( finite3207457112153483333omplex(B3)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
             => ? [Xa: complex] :
                  ( member_complex(Xa,B3)
                  & aa_complex_o(aa_nat_fun_complex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( member_complex(X2,B3)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_cz(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4393_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B3: set_nat,R: fun_int_fun_nat_o] :
      ( ~ finite_finite_int(A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X2: int] :
              ( member_int(X2,A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B3)
                  & aa_nat_o(aa_int_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B3)
              & ~ finite_finite_int(collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_da(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4394_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B3: set_int,R: fun_int_fun_int_o] :
      ( ~ finite_finite_int(A2)
     => ( finite_finite_int(B3)
       => ( ! [X2: int] :
              ( member_int(X2,A2)
             => ? [Xa: int] :
                  ( member_int(Xa,B3)
                  & aa_int_o(aa_int_fun_int_o(R,X2),Xa) ) )
         => ? [X2: int] :
              ( member_int(X2,B3)
              & ~ finite_finite_int(collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_db(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4395_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B3: set_complex,R: fun_in5065666251602127472plex_o] :
      ( ~ finite_finite_int(A2)
     => ( finite3207457112153483333omplex(B3)
       => ( ! [X2: int] :
              ( member_int(X2,A2)
             => ? [Xa: complex] :
                  ( member_complex(Xa,B3)
                  & aa_complex_o(aa_int_fun_complex_o(R,X2),Xa) ) )
         => ? [X2: complex] :
              ( member_complex(X2,B3)
              & ~ finite_finite_int(collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_dc(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4396_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B3: set_nat,R: fun_co5676208753217392844_nat_o] :
      ( ~ finite3207457112153483333omplex(A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( ! [X2: complex] :
              ( member_complex(X2,A2)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B3)
                  & aa_nat_o(aa_complex_fun_nat_o(R,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B3)
              & ~ finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_dd(A2),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_4397_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( finite_finite_real(A2)
     => ( member_real(A,A2)
       => ? [X2: real] :
            ( member_real(X2,A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),A)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4398_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(A,A2)
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & aa_set_nat_o(ord_less_eq_set_nat(X2),A)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4399_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ord_less_eq_set_int(X2,A)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( ord_less_eq_set_int(Xa,X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4400_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( finite_finite_rat(A2)
     => ( member_rat(A,A2)
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),A)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4401_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( finite_finite_num(A2)
     => ( member_num(A,A2)
       => ? [X2: num] :
            ( member_num(X2,A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),A)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4402_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4403_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( finite_finite_int(A2)
     => ( member_int(A,A2)
       => ? [X2: int] :
            ( member_int(X2,A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),A)
            & ! [Xa: int] :
                ( member_int(Xa,A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4404_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( finite_finite_real(A2)
     => ( member_real(A,A2)
       => ? [X2: real] :
            ( member_real(X2,A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4405_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( finite1152437895449049373et_nat(A2)
     => ( member_set_nat(A,A2)
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A2)
            & aa_set_nat_o(ord_less_eq_set_nat(A),X2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A2)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4406_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( member_set_int(A,A2)
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ord_less_eq_set_int(A,X2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( ord_less_eq_set_int(X2,Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4407_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( finite_finite_rat(A2)
     => ( member_rat(A,A2)
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),X2)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4408_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( finite_finite_num(A2)
     => ( member_num(A,A2)
       => ? [X2: num] :
            ( member_num(X2,A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),X2)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4409_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(member_nat(A),A2)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4410_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( finite_finite_int(A2)
     => ( member_int(A,A2)
       => ? [X2: int] :
            ( member_int(X2,A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X2)
            & ! [Xa: int] :
                ( member_int(Xa,A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4411_finite__subset,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% finite_subset
tff(fact_4412_finite__subset,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(A2,B3)
     => ( finite3207457112153483333omplex(B3)
       => finite3207457112153483333omplex(A2) ) ) ).

% finite_subset
tff(fact_4413_finite__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( finite_finite_int(B3)
       => finite_finite_int(A2) ) ) ).

% finite_subset
tff(fact_4414_infinite__super,axiom,
    ! [S3: set_nat,T2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S3)
       => ~ aa_set_nat_o(finite_finite_nat,T2) ) ) ).

% infinite_super
tff(fact_4415_infinite__super,axiom,
    ! [S3: set_complex,T2: set_complex] :
      ( ord_le211207098394363844omplex(S3,T2)
     => ( ~ finite3207457112153483333omplex(S3)
       => ~ finite3207457112153483333omplex(T2) ) ) ).

% infinite_super
tff(fact_4416_infinite__super,axiom,
    ! [S3: set_int,T2: set_int] :
      ( ord_less_eq_set_int(S3,T2)
     => ( ~ finite_finite_int(S3)
       => ~ finite_finite_int(T2) ) ) ).

% infinite_super
tff(fact_4417_rev__finite__subset,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => aa_set_nat_o(finite_finite_nat,A2) ) ) ).

% rev_finite_subset
tff(fact_4418_rev__finite__subset,axiom,
    ! [B3: set_complex,A2: set_complex] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => finite3207457112153483333omplex(A2) ) ) ).

% rev_finite_subset
tff(fact_4419_rev__finite__subset,axiom,
    ! [B3: set_int,A2: set_int] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(A2,B3)
       => finite_finite_int(A2) ) ) ).

% rev_finite_subset
tff(fact_4420_Diff__infinite__finite,axiom,
    ! [T2: set_int,S3: set_int] :
      ( finite_finite_int(T2)
     => ( ~ finite_finite_int(S3)
       => ~ finite_finite_int(minus_minus_set_int(S3,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_4421_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S3: set_complex] :
      ( finite3207457112153483333omplex(T2)
     => ( ~ finite3207457112153483333omplex(S3)
       => ~ finite3207457112153483333omplex(minus_811609699411566653omplex(S3,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_4422_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( ~ aa_set_nat_o(finite_finite_nat,S3)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S3,T2)) ) ) ).

% Diff_infinite_finite
tff(fact_4423_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( member_real(X2,A2)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4424_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( ord_less_eq_set_int(X2,Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4425_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( finite_finite_rat(A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4426_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( finite_finite_num(A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( member_num(X2,A2)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4427_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4428_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( finite_finite_int(A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( member_int(X2,A2)
            & ! [Xa: int] :
                ( member_int(Xa,A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4429_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ? [X2: real] :
            ( member_real(X2,A2)
            & ! [Xa: real] :
                ( member_real(Xa,A2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4430_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( finite6197958912794628473et_int(A2)
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X2: set_int] :
            ( member_set_int(X2,A2)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A2)
               => ( ord_less_eq_set_int(Xa,X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4431_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( finite_finite_rat(A2)
     => ( ( A2 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( member_rat(X2,A2)
            & ! [Xa: rat] :
                ( member_rat(Xa,A2)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4432_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( finite_finite_num(A2)
     => ( ( A2 != bot_bot_set_num )
       => ? [X2: num] :
            ( member_num(X2,A2)
            & ! [Xa: num] :
                ( member_num(Xa,A2)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4433_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A2)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4434_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( finite_finite_int(A2)
     => ( ( A2 != bot_bot_set_int )
       => ? [X2: int] :
            ( member_int(X2,A2)
            & ! [Xa: int] :
                ( member_int(Xa,A2)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4435_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4436_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_de(Nb),C))) ) ).

% finite_nth_roots
tff(fact_4437_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_df(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_df(I6),Y)))
       => finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_dg(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4438_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_dh(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_dh(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_di(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4439_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Y: fun_int_complex] :
      ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_dj(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_dj(I6),Y)))
       => finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_dk(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4440_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Y: fun_complex_complex] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_dl(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_dl(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dm(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4441_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dn(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dn(I6),Y)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_do(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4442_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_dp(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_dp(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dq(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4443_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dr(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dr(I6),Y)))
       => finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_ds(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4444_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_dt(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_dt(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_du(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4445_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_dv(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_dv(I6),Y)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_dw(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4446_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dx(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dx(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dy(I6),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_4447_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_dz(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_dz(I6),Y)))
       => finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ea(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4448_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_eb(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_eb(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_ec(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4449_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_complex,Y: fun_int_complex] :
      ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_ed(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_ed(I6),Y)))
       => finite_finite_int(collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ee(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4450_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_complex,Y: fun_complex_complex] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_ef(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_ef(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_eg(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4451_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_eh(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_eh(I6),Y)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ei(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4452_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ej(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ej(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_ek(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4453_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_el(I6),Xb)))
     => ( finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_el(I6),Y)))
       => finite_finite_int(collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_em(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4454_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_en(I6),Xb)))
     => ( finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_en(I6),Y)))
       => finite3207457112153483333omplex(collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_eo(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4455_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ep(I6),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_ep(I6),Y)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_eq(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4456_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_er(I6),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_er(I6),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_es(I6),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_4457_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M),zero_zero_nat)
     => ( ! [M: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_4458_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,Nb)),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_4459_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_4460_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_4461_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_4462_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_4463_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4464_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4465_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_4466_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_4467_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_4468_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_4469_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
    <=> ( Mb = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_4470_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_4471_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
    <=> ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_4472_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
    <=> ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_4473_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
    <=> ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_4474_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
    <=> ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_4475_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
    <=> ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_4476_dvd__0__right,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_4477_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),zero_zero_complex) ).

% dvd_0_right
tff(fact_4478_dvd__0__right,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),zero_zero_real) ).

% dvd_0_right
tff(fact_4479_dvd__0__right,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),zero_zero_rat) ).

% dvd_0_right
tff(fact_4480_dvd__0__right,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),zero_zero_nat) ).

% dvd_0_right
tff(fact_4481_dvd__0__right,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),zero_zero_int) ).

% dvd_0_right
tff(fact_4482_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4483_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4484_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4485_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4486_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_4487_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4488_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4489_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4490_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4491_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_4492_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_4493_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_4494_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_4495_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_4496_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_4497_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_4498_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).

% concat_bit_0
tff(fact_4499_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_4500_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_4501_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_4502_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_4503_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4504_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4505_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4506_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4507_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_4508_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4509_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4510_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4511_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4512_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_4513_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4514_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4515_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4516_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4517_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4518_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4519_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4520_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4521_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4522_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4523_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4524_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4525_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4526_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4527_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4528_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4529_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_4530_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_4531_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_4532_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_4533_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_4534_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_4535_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_4536_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_4537_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_4538_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_4539_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_4540_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_4541_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_4542_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_4543_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_4544_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_4545_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_4546_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_4547_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_4548_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_4549_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_4550_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_4551_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_4552_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_4553_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_4554_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_4555_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_4556_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_4557_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_4558_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_4559_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_4560_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_4561_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_4562_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_4563_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_4564_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_4565_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_4566_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_4567_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_4568_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_4569_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_4570_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_4571_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_4572_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_4573_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_4574_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_4575_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_4576_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_4577_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_4578_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_4579_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_4580_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_4581_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_4582_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_4583_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_4584_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_4585_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4586_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4587_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_4588_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4589_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4590_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_4591_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_4592_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_4593_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_4594_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_4595_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_4596_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_4597_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_4598_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_4599_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_4600_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_4601_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4602_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4603_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_4604_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4605_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4606_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_4607_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_4608_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_4609_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_4610_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4611_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4612_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_4613_even__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_4614_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_4615_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_4616_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_4617_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_4618_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_4619_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_4620_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4621_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4622_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_4623_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4624_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4625_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4626_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_4627_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_4628_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_trans
tff(fact_4629_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_4630_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_4631_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_4632_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B6: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C5) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B6),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),C) ) ) ).

% division_decomp
tff(fact_4633_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B6: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B6),C5) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B6),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),C) ) ) ).

% division_decomp
tff(fact_4634_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X2: nat,Y4: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Y4) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X2),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_4635_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X2: int,Y4: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X2),Y4) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X2),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_4636_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A)
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_4637_dvd__0__left,axiom,
    ! [A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,zero_zero_complex),A)
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_4638_dvd__0__left,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A)
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_4639_dvd__0__left,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A)
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_4640_dvd__0__left,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A)
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_4641_dvd__0__left,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A)
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_4642_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_4643_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)) ).

% dvd_triv_right
tff(fact_4644_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A)) ).

% dvd_triv_right
tff(fact_4645_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_4646_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_4647_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_4648_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_4649_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_4650_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_4651_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_4652_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4653_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4654_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4655_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4656_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_4657_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_4658_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% dvd_triv_left
tff(fact_4659_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B)) ).

% dvd_triv_left
tff(fact_4660_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_4661_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_4662_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_4663_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_4664_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_4665_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_4666_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_4667_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_4668_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult2
tff(fact_4669_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult2
tff(fact_4670_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_4671_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_4672_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_4673_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ).

% dvd_mult
tff(fact_4674_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult
tff(fact_4675_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_4676_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_4677_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvd_def
tff(fact_4678_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).

% dvd_def
tff(fact_4679_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_4680_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_4681_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_4682_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_4683_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_4684_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_4685_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_4686_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_4687_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvdE
tff(fact_4688_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).

% dvdE
tff(fact_4689_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K2: rat] : A != aa_rat_rat(times_times_rat(B),K2) ) ).

% dvdE
tff(fact_4690_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_4691_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvdE
tff(fact_4692_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_4693_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_4694_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_4695_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_4696_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_4697_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_4698_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_4699_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_4700_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_4701_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_4702_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)) ) ) ).

% dvd_add
tff(fact_4703_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ) ) ).

% dvd_add
tff(fact_4704_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)) ) ) ).

% dvd_add
tff(fact_4705_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_4706_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_4707_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_4708_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,one_one_complex),A) ).

% one_dvd
tff(fact_4709_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_4710_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_4711_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_4712_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_4713_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_4714_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_4715_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_4716_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_4717_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_4718_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_4719_dvd__diff,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),minus_8373710615458151222nteger(Y,Z)) ) ) ).

% dvd_diff
tff(fact_4720_dvd__diff,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ) ) ).

% dvd_diff
tff(fact_4721_dvd__diff,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_rat_rat(minus_minus_rat(Y),Z)) ) ) ).

% dvd_diff
tff(fact_4722_dvd__diff,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ) ) ).

% dvd_diff
tff(fact_4723_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B,C)) ) ).

% dvd_diff_commute
tff(fact_4724_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% dvd_diff_commute
tff(fact_4725_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_div_div_same
tff(fact_4726_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_4727_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_4728_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4729_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),A)
       => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4730_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4731_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4732_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4733_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_4734_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4735_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),A)
     => ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,C),B)
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),C) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4736_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4737_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4738_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4739_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_4740_dvd__power__same,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Nb)) ) ).

% dvd_power_same
tff(fact_4741_dvd__power__same,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).

% dvd_power_same
tff(fact_4742_dvd__power__same,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ) ).

% dvd_power_same
tff(fact_4743_dvd__power__same,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),Nb)) ) ).

% dvd_power_same
tff(fact_4744_dvd__power__same,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),Nb)) ) ).

% dvd_power_same
tff(fact_4745_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_4746_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) ) ) ).

% mod_mod_cancel
tff(fact_4747_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_4748_dvd__mod,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_4749_dvd__mod,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_4750_dvd__mod,axiom,
    ! [K: code_integer,Mb: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(Mb,Nb)) ) ) ).

% dvd_mod
tff(fact_4751_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4752_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4753_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_4754_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_4755_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_4756_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_4757_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_4758_zdvd__zdiffD,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),Nb))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_4759_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_4760_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [X2: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2) = 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)),Y4)),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)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_4761_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X2: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),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),Y4)),D3) )
        | ( 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),Y4)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_4762_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X2: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = D3 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_4763_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_le211207098394363844omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_et,A)),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_et,B)))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B) ) ).

% subset_divisors_dvd
tff(fact_4764_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eu,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eu,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_4765_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ev,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ev,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_4766_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ew,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_ew,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_4767_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Mb: nat,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),L),R2)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),R2) ).

% concat_bit_assoc
tff(fact_4768_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_less_set_complex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_et,A)),collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_et,B)))
    <=> ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,A),B)
        & ~ aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4769_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eu,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eu,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4770_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ew,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_ew,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4771_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ev,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_ev,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_4772_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero_int )
     => finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ew,I2))) ) ).

% finite_divisors_int
tff(fact_4773_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_4774_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_4775_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_4776_pinf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% pinf(9)
tff(fact_4777_pinf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% pinf(9)
tff(fact_4778_pinf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% pinf(9)
tff(fact_4779_pinf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% pinf(9)
tff(fact_4780_pinf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% pinf(9)
tff(fact_4781_pinf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% pinf(10)
tff(fact_4782_pinf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% pinf(10)
tff(fact_4783_pinf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% pinf(10)
tff(fact_4784_pinf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% pinf(10)
tff(fact_4785_pinf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% pinf(10)
tff(fact_4786_minf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% minf(9)
tff(fact_4787_minf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% minf(9)
tff(fact_4788_minf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% minf(9)
tff(fact_4789_minf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% minf(9)
tff(fact_4790_minf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% minf(9)
tff(fact_4791_minf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% minf(10)
tff(fact_4792_minf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),S2)) ) ) ).

% minf(10)
tff(fact_4793_minf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% minf(10)
tff(fact_4794_minf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% minf(10)
tff(fact_4795_minf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% minf(10)
tff(fact_4796_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4797_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4798_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4799_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4800_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4801_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_4802_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_4803_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_4804_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_4805_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4806_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4807_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4808_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4809_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4810_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4811_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4812_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4813_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4814_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4815_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4816_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4817_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4818_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4819_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4820_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4821_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4822_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4823_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4824_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4825_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_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_4826_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_4827_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_4828_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_4829_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4830_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4831_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4832_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4833_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4834_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4835_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_4836_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_4837_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_4838_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_4839_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_4840_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_4841_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_4842_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_4843_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_4844_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_4845_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_4846_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_4847_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_4848_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_4849_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_4850_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_4851_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_4852_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_4853_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4854_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4855_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4856_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_4857_div__power,axiom,
    ! [B: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_4858_div__power,axiom,
    ! [B: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% div_power
tff(fact_4859_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_4860_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_4861_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_4862_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4863_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4864_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_4865_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_4866_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_4867_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_4868_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_4869_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_4870_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_4871_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_4872_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_4873_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_4874_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_4875_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_4876_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_4877_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_4878_dvd__power__le,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_4879_dvd__power__le,axiom,
    ! [Xb: nat,Y: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_4880_dvd__power__le,axiom,
    ! [Xb: real,Y: real,Nb: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_4881_dvd__power__le,axiom,
    ! [Xb: int,Y: int,Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_4882_dvd__power__le,axiom,
    ! [Xb: complex,Y: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_4883_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_4884_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% dvd_minus_mod
tff(fact_4885_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_4886_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_4887_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B)) ) ).

% mod_eq_dvd_iff
tff(fact_4888_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X2: nat,Y4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D3),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),Y4)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_4889_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_4890_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_4891_zdvd__antisym__nonneg,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb)
           => ( Mb = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_4892_dvd__diffD,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb) ) ) ) ).

% dvd_diffD
tff(fact_4893_dvd__diffD1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_4894_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_4895_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_4896_zdvd__mono,axiom,
    ! [K: int,Mb: int,Ta: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Ta)) ) ) ).

% zdvd_mono
tff(fact_4897_dbl__def,axiom,
    ! [Xb: real] : aa_real_real(neg_numeral_dbl_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb) ).

% dbl_def
tff(fact_4898_dbl__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xb) = aa_rat_rat(plus_plus_rat(Xb),Xb) ).

% dbl_def
tff(fact_4899_dbl__def,axiom,
    ! [Xb: int] : aa_int_int(neg_numeral_dbl_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb) ).

% dbl_def
tff(fact_4900_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb) ) ).

% zdvd_reduce
tff(fact_4901_zdvd__period,axiom,
    ! [A: int,D: int,Xb: int,Ta: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Ta))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_4902_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eu,Mb))) ) ).

% finite_divisors_nat
tff(fact_4903_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_4904_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X3: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X3))
    <=> ? [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X3,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4905_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X3: complex] : aa_complex_o(P,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,L),X3))
    <=> ? [X3: complex] :
          ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,L),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X3),zero_zero_complex))
          & aa_complex_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4906_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X3: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X3))
    <=> ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),zero_zero_real))
          & aa_real_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4907_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X3: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X3))
    <=> ? [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X3),zero_zero_rat))
          & aa_rat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4908_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X3: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3))
    <=> ? [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat))
          & aa_nat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4909_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X3: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3))
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int))
          & aa_int_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_4910_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ).

% unit_dvdE
tff(fact_4911_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_4912_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_4913_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4914_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4915_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_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_4916_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4917_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4918_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4919_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4920_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4921_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4922_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4923_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4924_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_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_4925_even__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_4926_even__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_4927_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_4928_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4929_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4930_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4931_inf__period_I3_J,axiom,
    ! [D: code_integer,D2: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D2)
     => ! [X: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D2)),Ta)) ) ) ).

% inf_period(3)
tff(fact_4932_inf__period_I3_J,axiom,
    ! [D: real,D2: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D2)
     => ! [X: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4933_inf__period_I3_J,axiom,
    ! [D: rat,D2: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D2)
     => ! [X: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4934_inf__period_I3_J,axiom,
    ! [D: int,D2: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2))),Ta)) ) ) ).

% inf_period(3)
tff(fact_4935_inf__period_I4_J,axiom,
    ! [D: code_integer,D2: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D2)
     => ! [X: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D2)),Ta)) ) ) ).

% inf_period(4)
tff(fact_4936_inf__period_I4_J,axiom,
    ! [D: real,D2: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D2)
     => ! [X: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4937_inf__period_I4_J,axiom,
    ! [D: rat,D2: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D2)
     => ! [X: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4938_inf__period_I4_J,axiom,
    ! [D: int,D2: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D2))),Ta)) ) ) ).

% inf_period(4)
tff(fact_4939_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4940_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4941_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4942_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_4943_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4944_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4945_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_4946_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4947_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4948_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_4949_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4950_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4951_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4952_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4953_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_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_4954_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4955_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4956_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_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_4957_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4958_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4959_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4960_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_4961_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_4962_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_4963_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% dvd_imp_le
tff(fact_4964_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_4965_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_4966_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_4967_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_4968_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4969_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod_nat_nat] :
      ~ ! [K2: nat,M: nat] : Xb != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M) ).

% prod_decode_aux.cases
tff(fact_4970_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_4971_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_4972_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_4973_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_4974_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_4975_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_4976_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_4977_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_4978_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_4979_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4980_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4981_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4982_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4983_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4984_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4985_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B2 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B2) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4986_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B2 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = 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),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4987_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B2 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4988_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_4989_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_4990_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_4991_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B) )
        & ( divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4992_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4993_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_4994_dvd__power__iff,axiom,
    ! [Xb: code_integer,Mb: nat,Nb: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_4995_dvd__power__iff,axiom,
    ! [Xb: nat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_4996_dvd__power__iff,axiom,
    ! [Xb: int,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_4997_dvd__power,axiom,
    ! [Nb: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)) ) ).

% dvd_power
tff(fact_4998_dvd__power,axiom,
    ! [Nb: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ).

% dvd_power
tff(fact_4999_dvd__power,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).

% dvd_power
tff(fact_5000_dvd__power,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ).

% dvd_power
tff(fact_5001_dvd__power,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) ) ).

% dvd_power
tff(fact_5002_dvd__power,axiom,
    ! [Nb: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) ) ).

% dvd_power
tff(fact_5003_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_5004_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_5005_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_5006_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_5007_power__dvd__imp__le,axiom,
    ! [I2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_5008_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_5009_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_5010_bset_I9_J,axiom,
    ! [D: int,D2: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)),Ta)) ) ) ) ).

% bset(9)
tff(fact_5011_bset_I10_J,axiom,
    ! [D: int,D2: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D2)),Ta)) ) ) ) ).

% bset(10)
tff(fact_5012_aset_I9_J,axiom,
    ! [D: int,D2: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)),Ta)) ) ) ) ).

% aset(9)
tff(fact_5013_aset_I10_J,axiom,
    ! [D: int,D2: int,A2: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D2)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D2))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D2)),Ta)) ) ) ) ).

% aset(10)
tff(fact_5014_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5015_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5016_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_5017_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5018_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5019_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5020_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5021_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5022_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5023_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5024_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5025_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_5026_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_5027_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_5028_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5029_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5030_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5031_even__set__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5032_even__set__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5033_even__set__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5034_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5035_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Mb),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5036_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5037_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_diff_iff
tff(fact_5038_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
     => ~ ! [B2: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B2),one_one_Code_integer) ) ).

% oddE
tff(fact_5039_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B2: 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))),B2)),one_one_nat) ) ).

% oddE
tff(fact_5040_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B2: 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))),B2)),one_one_int) ) ).

% oddE
tff(fact_5041_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_5042_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_5043_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_5044_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_5045_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_5046_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_5047_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_even_power
tff(fact_5048_zero__le__even__power,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_5049_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_le_even_power
tff(fact_5050_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_5051_zero__le__odd__power,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_5052_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_5053_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_5054_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_5055_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_5056_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N: nat] : Xb != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_5057_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_5058_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_5059_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_5060_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B2),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),zero_zero_nat)
       => ( ! [A3: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_5061_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5062_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5063_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_5064_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5065_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5066_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5067_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_5068_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5069_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5070_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_5071_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5072_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5073_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5074_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X2: real] :
            ( member_real(X2,X5)
           => ? [Xa: real] :
                ( member_real(Xa,X5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Xa) ) )
       => ~ finite_finite_real(X5) ) ) ).

% infinite_growing
tff(fact_5075_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X2: rat] :
            ( member_rat(X2,X5)
           => ? [Xa: rat] :
                ( member_rat(Xa,X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Xa) ) )
       => ~ finite_finite_rat(X5) ) ) ).

% infinite_growing
tff(fact_5076_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X2: num] :
            ( member_num(X2,X5)
           => ? [Xa: num] :
                ( member_num(Xa,X5)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Xa) ) )
       => ~ finite_finite_num(X5) ) ) ).

% infinite_growing
tff(fact_5077_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),X5)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_5078_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( member_int(X2,X5)
           => ? [Xa: int] :
                ( member_int(Xa,X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Xa) ) )
       => ~ finite_finite_int(X5) ) ) ).

% infinite_growing
tff(fact_5079_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( finite_finite_real(S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X2: real] :
            ( member_real(X2,S3)
            & ~ ? [Xa: real] :
                  ( member_real(Xa,S3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5080_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( finite_finite_rat(S3)
     => ( ( S3 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( member_rat(X2,S3)
            & ~ ? [Xa: rat] :
                  ( member_rat(Xa,S3)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5081_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( finite_finite_num(S3)
     => ( ( S3 != bot_bot_set_num )
       => ? [X2: num] :
            ( member_num(X2,S3)
            & ~ ? [Xa: num] :
                  ( member_num(Xa,S3)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5082_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),S3)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S3)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5083_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
     => ( ( S3 != bot_bot_set_int )
       => ? [X2: int] :
            ( member_int(X2,S3)
            & ~ ? [Xa: int] :
                  ( member_int(Xa,S3)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5084_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_5085_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_5086_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_5087_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_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_5088_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_5089_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat,X22: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xb,some_P7363390416028606310at_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xb,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5090_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_nat_nat,X22: nat] : size_option_nat(Xb,some_nat(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Xb,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5091_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_num_nat,X22: num] : size_option_num(Xb,some_num(X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xb,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_5092_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Nb,divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5093_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5094_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),nat_set_decode(Xb))
    <=> aa_set_nat_o(member_nat(Nb),nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_5095_diff__shunt__var,axiom,
    ! [Xb: set_real,Y: set_real] :
      ( ( minus_minus_set_real(Xb,Y) = bot_bot_set_real )
    <=> ord_less_eq_set_real(Xb,Y) ) ).

% diff_shunt_var
tff(fact_5096_diff__shunt__var,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xb,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).

% diff_shunt_var
tff(fact_5097_diff__shunt__var,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xb,Y) = bot_bot_set_int )
    <=> ord_less_eq_set_int(Xb,Y) ) ).

% diff_shunt_var
tff(fact_5098_intind,axiom,
    ! [I2: nat,Nb: nat,P: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_o(P,Xb)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I2)) ) ) ).

% intind
tff(fact_5099_intind,axiom,
    ! [I2: nat,Nb: nat,P: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_int_o(P,Xb)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xb)),I2)) ) ) ).

% intind
tff(fact_5100_intind,axiom,
    ! [I2: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_VEBT_VEBT_o(P,Xb)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I2)) ) ) ).

% intind
tff(fact_5101_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5102_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5103_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5104_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_5105_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex($false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_5106_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_5107_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_5108_of__bool__eq_I1_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_5109_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_5110_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger($false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_5111_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_5112_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_5113_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_5114_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_5115_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = zero_zero_int )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_5116_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = zero_z3403309356797280102nteger )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_5117_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5118_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5119_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5120_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5121_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_5122_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5123_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5124_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5125_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5126_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5127_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_5128_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_5129_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_5130_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_5131_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_5132_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_5133_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_5134_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_5135_replicate__eq__replicate,axiom,
    ! [Mb: nat,Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Mb,Xb) = replicate_VEBT_VEBT(Nb,Y) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero_nat )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_5136_length__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_5137_length__replicate,axiom,
    ! [Nb: nat,Xb: $o] : size_size_list_o(replicate_o(Nb,(Xb))) = Nb ).

% length_replicate
tff(fact_5138_length__replicate,axiom,
    ! [Nb: nat,Xb: int] : size_size_list_int(replicate_int(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_5139_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        | (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_or_iff
tff(fact_5140_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        | (Q) )) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_or_iff
tff(fact_5141_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        | (Q) )) = ord_max_Code_integer(zero_n356916108424825756nteger((P)),zero_n356916108424825756nteger((Q))) ).

% of_bool_or_iff
tff(fact_5142_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5143_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5144_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5145_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5146_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_5147_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5148_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5149_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5150_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5151_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_5152_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n1201886186963655149omplex(~ (P)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),zero_n1201886186963655149omplex((P))) ).

% of_bool_not_iff
tff(fact_5153_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_5154_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_5155_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% of_bool_not_iff
tff(fact_5156_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n356916108424825756nteger(~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((P))) ).

% of_bool_not_iff
tff(fact_5157_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_5158_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_5159_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_5160_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5161_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_5162_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X3: int] :
          ( member_int(X3,set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5163_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_5164_in__set__replicate,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] :
      ( member_complex(Xb,set_complex2(replicate_complex(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5165_in__set__replicate,axiom,
    ! [Xb: real,Nb: nat,Y: real] :
      ( member_real(Xb,set_real2(replicate_real(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5166_in__set__replicate,axiom,
    ! [Xb: set_nat,Nb: nat,Y: set_nat] :
      ( member_set_nat(Xb,set_set_nat2(replicate_set_nat(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5167_in__set__replicate,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5168_in__set__replicate,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( member_int(Xb,set_int2(replicate_int(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5169_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Nb: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_5170_nth__replicate,axiom,
    ! [I2: nat,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I2) = Xb ) ) ).

% nth_replicate
tff(fact_5171_nth__replicate,axiom,
    ! [I2: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xb)),I2) = Xb ) ) ).

% nth_replicate
tff(fact_5172_nth__replicate,axiom,
    ! [I2: nat,Nb: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I2) = Xb ) ) ).

% nth_replicate
tff(fact_5173_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_5174_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_5175_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5176_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5177_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5178_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_5179_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_5180_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_5181_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),nat_set_decode(Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).

% set_decode_0
tff(fact_5182_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5183_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5184_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5185_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5186_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5187_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5188_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5189_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5190_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5191_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_5192_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = aa_o_nat(zero_n2687167440665602831ol_nat,(Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_5193_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P2)) = aa_o_int(zero_n2684676970156552555ol_int,(Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_5194_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n356916108424825756nteger((P2)) = zero_n356916108424825756nteger((Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_5195_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).

% of_bool_conj
tff(fact_5196_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2052037380579107095ol_rat(( (P)
        & (Q) )) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q))) ).

% of_bool_conj
tff(fact_5197_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q))) ).

% of_bool_conj
tff(fact_5198_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_5199_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q))) ).

% of_bool_conj
tff(fact_5200_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_5201_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_5202_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_5203_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P))) ).

% zero_less_eq_of_bool
tff(fact_5204_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P))) ).

% zero_less_eq_of_bool
tff(fact_5205_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P))) ).

% zero_less_eq_of_bool
tff(fact_5206_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% zero_less_eq_of_bool
tff(fact_5207_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P))) ).

% zero_less_eq_of_bool
tff(fact_5208_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_5209_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_5210_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_5211_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_5212_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_5213_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_5214_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_5215_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_5216_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_5217_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_5218_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_5219_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_5220_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_5221_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_5222_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_5223_split__of__bool,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ( ( (P2)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_5224_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ( ( (P2)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P2)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_5225_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_5226_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_5227_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_5228_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_5229_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P2)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_5230_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P2)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_5231_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_5232_replicate__length__same,axiom,
    ! [Xs: list_o,Xb: $o] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => ( (X2)
          <=> (Xb) ) )
     => ( replicate_o(size_size_list_o(Xs),(Xb)) = Xs ) ) ).

% replicate_length_same
tff(fact_5233_replicate__length__same,axiom,
    ! [Xs: list_int,Xb: int] :
      ( ! [X2: int] :
          ( member_int(X2,set_int2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_int(size_size_list_int(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_5234_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y4: complex] :
            ( member_complex(Y4,set_complex2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_complex(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5235_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y4: real] :
            ( member_real(Y4,set_real2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_real(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5236_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Nb: nat,Xb: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Nb )
     => ( ! [Y4: set_nat] :
            ( member_set_nat(Y4,set_set_nat2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_set_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5237_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y4: nat] :
            ( aa_set_nat_o(member_nat(Y4),set_nat2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5238_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y4: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y4,set_VEBT_VEBT2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5239_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y4: $o] :
            ( member_o((Y4),set_o2(Xs))
           => ( (Y4)
            <=> (Xb) ) )
       => ( Xs = replicate_o(Nb,(Xb)) ) ) ) ).

% replicate_eqI
tff(fact_5240_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y4: int] :
            ( member_int(Y4,set_int2(Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate_int(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_5241_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(nat_set_decode(Mb)),nat_set_decode(Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_5242_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5243_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] : aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5244_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A)) = modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_5245_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_5246_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Mb,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_5247_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Mb),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_5248_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => aa_nat_o(P,A3) )
     => ( ! [A3: nat,B2: $o] :
            ( aa_nat_o(P,A3)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_5249_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => aa_int_o(P,A3) )
     => ( ! [A3: int,B2: $o] :
            ( aa_int_o(P,A3)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_5250_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( divide6298287555418463151nteger(A3,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A3 )
         => aa_Code_integer_o(P,A3) )
     => ( ! [A3: code_integer,B2: $o] :
            ( aa_Code_integer_o(P,A3)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A3)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = A3 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A3))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_5251_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_5252_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_5253_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5254_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5255_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5256_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5257_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_nat_nat] : size_option_nat(Xb,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5258_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(Xb,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5259_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_num_nat] : size_option_num(Xb,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_5260_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ex,Xb)) ).

% set_decode_def
tff(fact_5261_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5262_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5263_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Mb) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5264_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_5265_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_5266_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5267_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5268_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5269_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(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),C)) != aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5270_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_5271_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_5272_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_5273_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_5274_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = 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),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5275_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_5276_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ey(L)),Qr) ).

% divmod_step_def
tff(fact_5277_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ez(L)),Qr) ).

% divmod_step_def
tff(fact_5278_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_fa(L),Qr) ).

% divmod_step_def
tff(fact_5279_Compl__anti__mono,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ord_less_eq_set_int(uminus1532241313380277803et_int(B3),uminus1532241313380277803et_int(A2)) ) ).

% Compl_anti_mono
tff(fact_5280_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(uminus1532241313380277803et_int(A2),uminus1532241313380277803et_int(B3))
    <=> ord_less_eq_set_int(B3,A2) ) ).

% Compl_subset_Compl_iff
tff(fact_5281_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_5282_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_5283_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_5284_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_5285_compl__le__compl__iff,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_eq_set_int(uminus1532241313380277803et_int(Xb),uminus1532241313380277803et_int(Y))
    <=> ord_less_eq_set_int(Y,Xb) ) ).

% compl_le_compl_iff
tff(fact_5286_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_5287_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_5288_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_5289_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_5290_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5291_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5292_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5293_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5294_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_5295_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_5296_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_5297_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) ).

% mult_minus_right
tff(fact_5298_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_5299_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_5300_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_5301_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_5302_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B) ).

% minus_mult_minus
tff(fact_5303_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_5304_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_5305_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_5306_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_5307_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) ).

% mult_minus_left
tff(fact_5308_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_5309_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_5310_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_5311_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_5312_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_5313_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_5314_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_5315_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_5316_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_5317_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = B ).

% minus_add_cancel
tff(fact_5318_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_5319_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_5320_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_5321_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_5322_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_5323_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_5324_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_5325_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_5326_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_5327_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),A) ).

% minus_diff_eq
tff(fact_5328_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_5329_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_5330_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% div_minus_minus
tff(fact_5331_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_5332_minus__dvd__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xb)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_5333_minus__dvd__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xb)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_5334_minus__dvd__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_5335_minus__dvd__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,Xb)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_5336_minus__dvd__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_5337_dvd__minus__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_5338_dvd__minus__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_5339_dvd__minus__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_complex_complex(uminus1482373934393186551omplex,Y))
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_5340_dvd__minus__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_rat_rat(uminus_uminus_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_5341_dvd__minus__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_5342_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_minus_minus
tff(fact_5343_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_5344_real__add__minus__iff,axiom,
    ! [Xb: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xb = A ) ) ).

% real_add_minus_iff
tff(fact_5345_sum_Oneutral__const,axiom,
    ! [A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_bm,A2) = zero_zero_int ).

% sum.neutral_const
tff(fact_5346_sum_Oneutral__const,axiom,
    ! [A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_bi,A2) = zero_zero_complex ).

% sum.neutral_const
tff(fact_5347_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bl),A2) = zero_zero_nat ).

% sum.neutral_const
tff(fact_5348_sum_Oneutral__const,axiom,
    ! [A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fb),A2) = zero_zero_real ).

% sum.neutral_const
tff(fact_5349_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_5350_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_5351_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_5352_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_5353_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_5354_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_5355_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_5356_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_5357_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_5358_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_5359_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_5360_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_5361_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_5362_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_5363_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_5364_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_5365_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_5366_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_5367_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_5368_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_5369_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_5370_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_5371_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_5372_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_5373_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_5374_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_5375_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_5376_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_5377_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_5378_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_5379_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_5380_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_5381_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_5382_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_5383_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_5384_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_5385_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_5386_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_5387_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_5388_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_5389_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_5390_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_5391_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_5392_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_5393_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_5394_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_5395_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_5396_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_5397_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_5398_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_5399_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_5400_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_5401_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_5402_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_5403_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_5404_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_5405_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_5406_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5407_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5408_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5409_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5410_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_5411_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_5412_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_5413_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_5414_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_5415_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_5416_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_5417_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_5418_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_5419_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_5420_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_5421_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_5422_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_5423_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) ).

% diff_minus_eq_add
tff(fact_5424_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_5425_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_5426_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_5427_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_5428_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),A) ).

% uminus_add_conv_diff
tff(fact_5429_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_5430_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_5431_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_5432_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_5433_divide__minus1,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xb) ).

% divide_minus1
tff(fact_5434_divide__minus1,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% divide_minus1
tff(fact_5435_divide__minus1,axiom,
    ! [Xb: rat] : aa_rat_rat(divide_divide_rat(Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xb) ).

% divide_minus1
tff(fact_5436_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_5437_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_mod_self1
tff(fact_5438_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5439_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5440_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_5441_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_5442_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_fc(A),B),S3) = $ite(member_real(A,S3),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_5443_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_fd(A),B)),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_5444_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( finite_finite_int(S3)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_fe(A),B),S3) = $ite(member_int(A,S3),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta'
tff(fact_5445_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ff(A),B),S3) = $ite(member_real(A,S3),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_5446_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( finite_finite_int(S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fg(A),B),S3) = $ite(member_int(A,S3),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_5447_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( finite3207457112153483333omplex(S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fh(A),B),S3) = $ite(member_complex(A,S3),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_5448_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fi(A),B),S3) = $ite(member_real(A,S3),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_5449_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fj(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_5450_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fk(A),B),S3) = $ite(member_int(A,S3),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_5451_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat] :
      ( finite3207457112153483333omplex(S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fl(A),B),S3) = $ite(member_complex(A,S3),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_5452_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_complex] :
      ( finite_finite_real(S3)
     => ( groups5754745047067104278omplex(aa_fun2233226938935547116omplex(aTP_Lamm_fm(A),B),S3) = $ite(member_real(A,S3),aa_real_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_5453_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_fn(A),B)),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_5454_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_complex] :
      ( finite_finite_int(S3)
     => ( groups3049146728041665814omplex(aa_fun3088484340408535532omplex(aTP_Lamm_fo(A),B),S3) = $ite(member_int(A,S3),aa_int_complex(B,A),zero_zero_complex) ) ) ).

% sum.delta
tff(fact_5455_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_real] :
      ( finite_finite_real(S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fp(A),B),S3) = $ite(member_real(A,S3),aa_real_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_5456_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_real] :
      ( finite_finite_int(S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fq(A),B),S3) = $ite(member_int(A,S3),aa_int_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_5457_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_real] :
      ( finite3207457112153483333omplex(S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fr(A),B),S3) = $ite(member_complex(A,S3),aa_complex_real(B,A),zero_zero_real) ) ) ).

% sum.delta
tff(fact_5458_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: fun_real_rat] :
      ( finite_finite_real(S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fs(A),B),S3) = $ite(member_real(A,S3),aa_real_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_5459_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ft(A),B),S3) = $ite(aa_set_nat_o(member_nat(A),S3),aa_nat_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_5460_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: fun_int_rat] :
      ( finite_finite_int(S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fu(A),B),S3) = $ite(member_int(A,S3),aa_int_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_5461_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: fun_complex_rat] :
      ( finite3207457112153483333omplex(S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fv(A),B),S3) = $ite(member_complex(A,S3),aa_complex_rat(B,A),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_5462_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_5463_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_5464_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_5465_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_5466_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_simps(1)
tff(fact_5467_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_5468_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_5469_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_5470_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_5471_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_5472_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_5473_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_5474_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_5475_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_5476_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_5477_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_5478_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_5479_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(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_5480_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_5481_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_5482_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5483_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5484_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5485_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5486_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_5487_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5488_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5489_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5490_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5491_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_5492_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5493_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5494_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5495_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5496_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_5497_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_5498_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_5499_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_5500_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_5501_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_5502_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_5503_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5504_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      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))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),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_5505_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(4)
tff(fact_5506_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      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))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),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_5507_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),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_5508_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      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)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)),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_5509_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(3)
tff(fact_5510_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      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)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),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_5511_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),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_5512_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      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))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),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_5513_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(2)
tff(fact_5514_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      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))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),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_5515_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),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_5516_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_5517_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_5518_semiring__norm_I168_J,axiom,
    ! [V: num,W: 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,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5519_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5520_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5521_semiring__norm_I168_J,axiom,
    ! [V: num,W: 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,W))),Y)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_5522_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_5523_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5524_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5525_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5526_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5527_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_5528_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5529_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5530_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5531_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5532_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_5533_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5534_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5535_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5536_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5537_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_5538_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5539_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5540_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5541_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5542_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_5543_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5544_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5545_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5546_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5547_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_5548_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5549_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5550_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5551_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5552_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_5553_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: 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_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5554_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: 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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5555_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5556_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5557_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_5558_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5559_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5560_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5561_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5562_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_5563_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5564_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5565_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5566_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_5567_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5568_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5569_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5570_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_5571_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5572_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5573_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5574_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_5575_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5576_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_5577_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5578_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_5579_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5580_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5581_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_5582_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: 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,W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != 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,W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5583_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5584_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_5585_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5586_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5587_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5588_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_5589_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5590_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_5591_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5592_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_5593_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5594_power2__minus,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5595_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5596_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5597_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5598_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_5599_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_5600_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(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_5601_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_5602_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5603_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_5604_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_5605_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(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_5606_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_5607_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5608_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_5609_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_5610_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(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_5611_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_5612_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5613_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_5614_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_5615_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_5616_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_5617_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_5618_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_5619_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5620_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5621_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5622_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5623_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5624_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5625_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5626_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5627_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5628_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5629_power__minus__odd,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5630_power__minus__odd,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5631_power__minus__odd,axiom,
    ! [Nb: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5632_power__minus__odd,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5633_power__minus__odd,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5634_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5635_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5636_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5637_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5638_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5639_diff__numeral__special_I3_J,axiom,
    ! [Nb: 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,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5640_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5641_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5642_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5643_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5644_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5645_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_5646_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_5647_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_5648_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_5649_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5650_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_5651_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_5652_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_5653_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_5654_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_5655_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_5656_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_5657_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_5658_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_5659_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_5660_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_5661_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_5662_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_5663_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_5664_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_5665_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5666_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5667_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K))) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_5668_sum__negf,axiom,
    ! [F: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aTP_Lamm_fw(F),A2) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(F,A2)) ).

% sum_negf
tff(fact_5669_sum__negf,axiom,
    ! [F: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_fx(F),A2) = aa_complex_complex(uminus1482373934393186551omplex,groups7754918857620584856omplex(F,A2)) ).

% sum_negf
tff(fact_5670_sum__negf,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fy(F)),A2) = aa_real_real(uminus_uminus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)) ).

% sum_negf
tff(fact_5671_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B3: set_int,A2: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_fz(G),B3),A2) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_gb(G),A2),B3) ).

% sum.swap
tff(fact_5672_sum_Oswap,axiom,
    ! [G: fun_co7673397065115134084omplex,B3: set_complex,A2: set_complex] : groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_gc(G),B3),A2) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_ge(G),A2),B3) ).

% sum.swap
tff(fact_5673_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B3: set_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gf(G),B3)),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gh(G),A2)),B3) ).

% sum.swap
tff(fact_5674_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_real,B3: set_nat,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_gi(G),B3)),A2) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_gk(G),A2)),B3) ).

% sum.swap
tff(fact_5675_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_gl(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5676_prod_Ocase__distrib,axiom,
    ! [H: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7323606162024495668nt_int(aTP_Lamm_gm(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5677_prod_Ocase__distrib,axiom,
    ! [H: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_gn(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5678_prod_Ocase__distrib,axiom,
    ! [H: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun6119123525526402106nt_int(aTP_Lamm_go(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5679_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_gp(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5680_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7461304428735181397nt_int(aTP_Lamm_gq(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5681_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_gr(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5682_prod_Ocase__distrib,axiom,
    ! [H: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_gs(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5683_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_gt(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5684_prod_Ocase__distrib,axiom,
    ! [H: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_gu(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5685_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_5686_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_5687_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_5688_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_5689_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_5690_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_5691_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_5692_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_5693_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_5694_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_5695_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_5696_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_5697_compl__le__swap2,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( ord_less_eq_set_int(uminus1532241313380277803et_int(Y),Xb)
     => ord_less_eq_set_int(uminus1532241313380277803et_int(Xb),Y) ) ).

% compl_le_swap2
tff(fact_5698_compl__le__swap1,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( ord_less_eq_set_int(Y,uminus1532241313380277803et_int(Xb))
     => ord_less_eq_set_int(Xb,uminus1532241313380277803et_int(Y)) ) ).

% compl_le_swap1
tff(fact_5699_compl__mono,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_less_eq_set_int(Xb,Y)
     => ord_less_eq_set_int(uminus1532241313380277803et_int(Y),uminus1532241313380277803et_int(Xb)) ) ).

% compl_mono
tff(fact_5700_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_5701_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_5702_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_5703_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_5704_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_5705_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_5706_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_5707_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_5708_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5709_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5710_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5711_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5712_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_real(numeral_numeral_real,Mb) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5713_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,Mb) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5714_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6690914467698888265omplex(Mb) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5715_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Mb) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5716_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_Code_integer(numera6620942414471956472nteger,Mb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5717_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5718_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5719_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5720_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5721_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)) != aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5722_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_5723_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_5724_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_5725_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_5726_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_5727_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_5728_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_5729_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_5730_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_5731_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_5732_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_5733_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_5734_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_5735_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_5736_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_5737_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_5738_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_5739_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_5740_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_5741_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_5742_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_5743_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_5744_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_5745_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_5746_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_5747_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_5748_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_5749_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_5750_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_5751_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_5752_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% minus_diff_minus
tff(fact_5753_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ).

% minus_diff_minus
tff(fact_5754_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) ).

% minus_diff_minus
tff(fact_5755_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) ).

% minus_diff_minus
tff(fact_5756_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_5757_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_5758_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_5759_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_5760_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_5761_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_5762_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_5763_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% div_minus_right
tff(fact_5764_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_5765_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_5766_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_5767_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_5768_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) ).

% minus_divide_divide
tff(fact_5769_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_5770_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_5771_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_5772_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_5773_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X22: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X22) ).

% old.prod.case
tff(fact_5774_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X22: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X22)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X22) ).

% old.prod.case
tff(fact_5775_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X22: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X22) ).

% old.prod.case
tff(fact_5776_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X22: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X22) ) ).

% old.prod.case
tff(fact_5777_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X22: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int_int(aa_int_fun_int_int(F,X1),X22) ).

% old.prod.case
tff(fact_5778_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_5779_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).

% mod_minus_right
tff(fact_5780_mod__minus__cong,axiom,
    ! [A: int,B: int,A5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),B) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A5)),B) ) ) ).

% mod_minus_cong
tff(fact_5781_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A5: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A5,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A5),B) ) ) ).

% mod_minus_cong
tff(fact_5782_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_5783_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% mod_minus_eq
tff(fact_5784_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I3: complex] :
          ( member_complex(I3,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5)) ) ).

% sum_mono
tff(fact_5785_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( member_real(I3,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5)) ) ).

% sum_mono
tff(fact_5786_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5)) ) ).

% sum_mono
tff(fact_5787_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( member_int(I3,K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_5788_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I3: complex] :
          ( member_complex(I3,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5)) ) ).

% sum_mono
tff(fact_5789_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I3: real] :
          ( member_real(I3,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_5790_sum__mono,axiom,
    ! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I3: int] :
          ( member_int(I3,K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_5791_sum__mono,axiom,
    ! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( ! [I3: complex] :
          ( member_complex(I3,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I3)),aa_complex_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5)) ) ).

% sum_mono
tff(fact_5792_sum__mono,axiom,
    ! [K5: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I3: real] :
          ( member_real(I3,K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_5793_sum__mono,axiom,
    ! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5)) ) ).

% sum_mono
tff(fact_5794_sum__product,axiom,
    ! [F: fun_int_int,A2: set_int,G: fun_int_int,B3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,B3)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_gw(F),G),B3),A2) ).

% sum_product
tff(fact_5795_sum__product,axiom,
    ! [F: fun_complex_complex,A2: set_complex,G: fun_complex_complex,B3: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,B3)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_gy(F),G),B3),A2) ).

% sum_product
tff(fact_5796_sum__product,axiom,
    ! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_ha(F),G),B3)),A2) ).

% sum_product
tff(fact_5797_sum__product,axiom,
    ! [F: fun_nat_real,A2: set_nat,G: fun_nat_real,B3: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(G),B3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_hc(F),G),B3)),A2) ).

% sum_product
tff(fact_5798_sum__distrib__right,axiom,
    ! [F: fun_int_int,A2: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_hd(F),R2),A2) ).

% sum_distrib_right
tff(fact_5799_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_he(F),R2),A2) ).

% sum_distrib_right
tff(fact_5800_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A2: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hf(F),R2)),A2) ).

% sum_distrib_right
tff(fact_5801_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),R2) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hg(F),R2)),A2) ).

% sum_distrib_right
tff(fact_5802_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A2: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hh(R2),F),A2) ).

% sum_distrib_left
tff(fact_5803_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_hi(R2),F),A2) ).

% sum_distrib_left
tff(fact_5804_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hj(R2),F)),A2) ).

% sum_distrib_left
tff(fact_5805_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_set_nat_real(groups6591440286371151544t_real(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(R2),F)),A2) ).

% sum_distrib_left
tff(fact_5806_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hl(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_5807_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_hm(G),H),A2) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).

% sum.distrib
tff(fact_5808_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_hn(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_5809_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_ho(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_5810_sum__subtractf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hp(F),G),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ).

% sum_subtractf
tff(fact_5811_sum__subtractf,axiom,
    ! [F: fun_complex_complex,G: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_hq(F),G),A2) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,A2)) ).

% sum_subtractf
tff(fact_5812_sum__subtractf,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,A2: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_hr(F),G)),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(G),A2)) ).

% sum_subtractf
tff(fact_5813_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A2: set_complex,R2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups7754918857620584856omplex(F,A2)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_hs(F),R2),A2) ).

% sum_divide_distrib
tff(fact_5814_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A2: set_nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),R2) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ht(F),R2)),A2) ).

% sum_divide_distrib
tff(fact_5815_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_int,G: fun_real_fun_int_int,R: fun_real_fun_int_o] :
      ( finite_finite_real(A2)
     => ( finite_finite_int(B3)
       => ( groups1932886352136224148al_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_hv(B3),G),R),A2) = groups4538972089207619220nt_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_hx(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5816_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite_finite_int(B3)
       => ( groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_hz(B3),G),R),A2) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_ib(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5817_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_int,G: fun_co333780937270390912nt_int,R: fun_co3431063785546946928_int_o] :
      ( finite3207457112153483333omplex(A2)
     => ( finite_finite_int(B3)
       => ( groups5690904116761175830ex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_id(B3),G),R),A2) = groups4538972089207619220nt_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_ig(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5818_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_complex,G: fun_re7208826350674946562omplex,R: fun_re7319081385389557872plex_o] :
      ( finite_finite_real(A2)
     => ( finite3207457112153483333omplex(B3)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_ii(B3),G),R),A2) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_ik(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5819_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B3: set_complex,G: fun_na1591161823130732966omplex,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( finite3207457112153483333omplex(B3)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_im(B3),G),R)),A2) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_io(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5820_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B3: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5065666251602127472plex_o] :
      ( finite_finite_int(A2)
     => ( finite3207457112153483333omplex(B3)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_iq(B3),G),R),A2) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_is(A2),G),R),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5821_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_iu(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_iw(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5822_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B3: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( finite_finite_int(A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_iy(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ja(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5823_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B3: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( finite3207457112153483333omplex(A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_jc(B3),G),R),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_je(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5824_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B3: set_nat,G: fun_re4238438383274954274t_real,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A2)
     => ( aa_set_nat_o(finite_finite_nat,B3)
       => ( groups8097168146408367636l_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_jf(B3),G),R),A2) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_jh(A2),G),R)),B3) ) ) ) ).

% sum.swap_restrict
tff(fact_5825_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ji(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A2)),A) ).

% mod_sum_eq
tff(fact_5826_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jj(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).

% mod_sum_eq
tff(fact_5827_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X2: nat,Y4: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X2),Y4) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5828_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X2: nat,Y4: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X2),Y4) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4))
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5829_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X2: int,Y4: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X2),Y4) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5830_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: int,Y4: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X2),Y4)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5831_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X2: int,Y4: int] : aa_int_int(aa_int_fun_int_int(F,X2),Y4) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4))
     => ( produc8211389475949308722nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5832_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_jk(F)) = F ).

% case_prod_eta
tff(fact_5833_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_jl(F)) = F ).

% case_prod_eta
tff(fact_5834_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_jm(F)) = F ).

% case_prod_eta
tff(fact_5835_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_jn(F)) = F ).

% case_prod_eta
tff(fact_5836_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(aTP_Lamm_jo(F)) = F ).

% case_prod_eta
tff(fact_5837_case__prodE2,axiom,
    ! [Q: fun_fu8391177546919232046_nat_o,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z))
     => ~ ! [X2: nat,Y4: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5838_case__prodE2,axiom,
    ! [Q: fun_fu6891447142836604759at_o_o,P: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(P),Z))
     => ~ ! [X2: nat,Y4: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5839_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
     => ~ ! [X2: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5840_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
     => ~ ! [X2: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5841_case__prodE2,axiom,
    ! [Q: fun_int_o,P: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(P),Z))
     => ~ ! [X2: int,Y4: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_5842_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A2)) ) ).

% sum_nonneg
tff(fact_5843_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2)) ) ).

% sum_nonneg
tff(fact_5844_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2)) ) ).

% sum_nonneg
tff(fact_5845_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_5846_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_5847_sum__nonneg,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_5848_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2)) ) ).

% sum_nonneg
tff(fact_5849_sum__nonneg,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_5850_sum__nonneg,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_5851_sum__nonneg,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2)) ) ).

% sum_nonneg
tff(fact_5852_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_5853_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_5854_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_5855_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5856_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5857_sum__nonpos,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5858_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_5859_sum__nonpos,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5860_sum__nonpos,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5861_sum__nonpos,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_5862_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_5863_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I6: set_real,G: fun_real_rat,I2: real] :
      ( ( groups1300246762558778688al_rat(F,I6) = groups1300246762558778688al_rat(G,I6) )
     => ( ! [I3: real] :
            ( member_real(I3,I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) )
       => ( member_real(I2,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_rat(F,I2) = aa_real_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5864_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I6: set_nat,G: fun_nat_rat,I2: nat] :
      ( ( groups2906978787729119204at_rat(F,I6) = groups2906978787729119204at_rat(G,I6) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) )
       => ( aa_set_nat_o(member_nat(I2),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_rat(F,I2) = aa_nat_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5865_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I6: set_int,G: fun_int_rat,I2: int] :
      ( ( groups3906332499630173760nt_rat(F,I6) = groups3906332499630173760nt_rat(G,I6) )
     => ( ! [I3: int] :
            ( member_int(I3,I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) )
       => ( member_int(I2,I6)
         => ( finite_finite_int(I6)
           => ( aa_int_rat(F,I2) = aa_int_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5866_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I6: set_complex,G: fun_complex_rat,I2: complex] :
      ( ( groups5058264527183730370ex_rat(F,I6) = groups5058264527183730370ex_rat(G,I6) )
     => ( ! [I3: complex] :
            ( member_complex(I3,I6)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) )
       => ( member_complex(I2,I6)
         => ( finite3207457112153483333omplex(I6)
           => ( aa_complex_rat(F,I2) = aa_complex_rat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5867_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I6: set_real,G: fun_real_nat,I2: real] :
      ( ( groups1935376822645274424al_nat(F,I6) = groups1935376822645274424al_nat(G,I6) )
     => ( ! [I3: real] :
            ( member_real(I3,I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I3)),aa_real_nat(G,I3)) )
       => ( member_real(I2,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_nat(F,I2) = aa_real_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5868_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I6: set_int,G: fun_int_nat,I2: int] :
      ( ( groups4541462559716669496nt_nat(F,I6) = groups4541462559716669496nt_nat(G,I6) )
     => ( ! [I3: int] :
            ( member_int(I3,I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I3)),aa_int_nat(G,I3)) )
       => ( member_int(I2,I6)
         => ( finite_finite_int(I6)
           => ( aa_int_nat(F,I2) = aa_int_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5869_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I6: set_complex,G: fun_complex_nat,I2: complex] :
      ( ( groups5693394587270226106ex_nat(F,I6) = groups5693394587270226106ex_nat(G,I6) )
     => ( ! [I3: complex] :
            ( member_complex(I3,I6)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I3)),aa_complex_nat(G,I3)) )
       => ( member_complex(I2,I6)
         => ( finite3207457112153483333omplex(I6)
           => ( aa_complex_nat(F,I2) = aa_complex_nat(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5870_sum__mono__inv,axiom,
    ! [F: fun_real_int,I6: set_real,G: fun_real_int,I2: real] :
      ( ( groups1932886352136224148al_int(F,I6) = groups1932886352136224148al_int(G,I6) )
     => ( ! [I3: real] :
            ( member_real(I3,I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I3)),aa_real_int(G,I3)) )
       => ( member_real(I2,I6)
         => ( finite_finite_real(I6)
           => ( aa_real_int(F,I2) = aa_real_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5871_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I6: set_nat,G: fun_nat_int,I2: nat] :
      ( ( groups3539618377306564664at_int(F,I6) = groups3539618377306564664at_int(G,I6) )
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I3)),aa_nat_int(G,I3)) )
       => ( aa_set_nat_o(member_nat(I2),I6)
         => ( aa_set_nat_o(finite_finite_nat,I6)
           => ( aa_nat_int(F,I2) = aa_nat_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5872_sum__mono__inv,axiom,
    ! [F: fun_complex_int,I6: set_complex,G: fun_complex_int,I2: complex] :
      ( ( groups5690904116761175830ex_int(F,I6) = groups5690904116761175830ex_int(G,I6) )
     => ( ! [I3: complex] :
            ( member_complex(I3,I6)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I3)),aa_complex_int(G,I3)) )
       => ( member_complex(I2,I6)
         => ( finite3207457112153483333omplex(I6)
           => ( aa_complex_int(F,I2) = aa_complex_int(G,I2) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5873_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5874_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5875_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5876_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5877_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5878_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5879_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5880_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5881_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5882_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5883_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5884_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5885_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5886_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5887_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5888_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5889_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5890_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5891_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5892_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5893_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5894_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_5895_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_5896_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_5897_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_5898_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_5899_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_5900_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_5901_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_5902_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_5903_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_5904_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_5905_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_5906_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_5907_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_5908_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_5909_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_5910_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_5911_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5912_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_5913_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_5914_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_5915_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_5916_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5917_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_5918_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_5919_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_5920_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_5921_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_5922_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_5923_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_5924_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_5925_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_5926_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_5927_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5928_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5929_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5930_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5931_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5932_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_5933_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_5934_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_5935_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xb)) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_5936_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,W))) ).

% numeral_times_minus_swap
tff(fact_5937_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_5938_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_5939_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_5940_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_5941_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_5942_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_5943_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_5944_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_5945_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_5946_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5947_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_5948_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_5949_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5950_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_5951_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5952_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_5953_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5954_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5955_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_Code_integer(numera6620942414471956472nteger,Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5956_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% one_neq_neg_numeral
tff(fact_5957_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_5958_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_5959_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_neq_neg_numeral
tff(fact_5960_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb)) ).

% one_neq_neg_numeral
tff(fact_5961_square__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb) = one_one_real )
    <=> ( ( Xb = one_one_real )
        | ( Xb = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_5962_square__eq__1__iff,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb) = one_one_int )
    <=> ( ( Xb = one_one_int )
        | ( Xb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_5963_square__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb) = one_one_complex )
    <=> ( ( Xb = one_one_complex )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_5964_square__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Xb) = one_one_rat )
    <=> ( ( Xb = one_one_rat )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_5965_square__eq__1__iff,axiom,
    ! [Xb: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xb) = one_one_Code_integer )
    <=> ( ( Xb = one_one_Code_integer )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_5966_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B3) = 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_5967_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3 = 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),B3) = 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_5968_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,K),B) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B3) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5969_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B3) = 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_5970_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5971_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_5972_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_5973_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_5974_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_5975_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_5976_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_5977_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_5978_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5979_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_5980_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_5981_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5982_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5983_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5984_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5985_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_5986_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5987_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5988_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,B),A)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5989_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5990_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_5991_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),uminus5710092332889474511et_nat(A2))
    <=> ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5992_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,uminus612125837232591019t_real(A2))
    <=> ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_5993_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ord_less_eq_set_int(A2,uminus1532241313380277803et_int(A2))
    <=> ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_5994_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_5995_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
     => ( ( Mb = one_one_int )
        | ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5996_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5997_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_5998_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_complex,P: fun_real_o] :
      ( finite_finite_real(A2)
     => ( groups5754745047067104278omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A2),P))) = groups5754745047067104278omplex(aa_fun1098856492951973088omplex(aTP_Lamm_jp(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_5999_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_complex,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(A2),P))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun1400689748890728928omplex(aTP_Lamm_jq(G),P)),A2) ) ) ).

% sum.inter_filter
tff(fact_6000_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_complex,P: fun_int_o] :
      ( finite_finite_int(A2)
     => ( groups3049146728041665814omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A2),P))) = groups3049146728041665814omplex(aa_fun6439487526662521568omplex(aTP_Lamm_jr(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6001_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_real,P: fun_real_o] :
      ( finite_finite_real(A2)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A2),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_js(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6002_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_real,P: fun_int_o] :
      ( finite_finite_int(A2)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A2),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_jt(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6003_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A2),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_ju(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6004_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: fun_real_rat,P: fun_real_o] :
      ( finite_finite_real(A2)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bc(A2),P))) = groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_jv(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6005_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(A2),P))) = groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_jw(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6006_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: fun_int_rat,P: fun_int_o] :
      ( finite_finite_int(A2)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A2),P))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_jx(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6007_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: fun_complex_rat,P: fun_complex_o] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_be(A2),P))) = groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_jy(G),P),A2) ) ) ).

% sum.inter_filter
tff(fact_6008_minus__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_6009_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_6010_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_int,G: fun_int_real,I2: fun_int_int,F: fun_int_real] :
      ( finite_finite_int(S2)
     => ( finite_finite_int(Ta)
       => ( ! [X2: int] :
              ( member_int(X2,Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: int] :
                ( member_int(X2,S2)
               => ? [Xa: int] :
                    ( member_int(Xa,Ta)
                    & ( aa_int_int(I2,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6011_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_complex,G: fun_complex_real,I2: fun_complex_int,F: fun_int_real] :
      ( finite_finite_int(S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X2: complex] :
              ( member_complex(X2,Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: int] :
                ( member_int(X2,S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_int(I2,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S2)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6012_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_int,G: fun_int_real,I2: fun_int_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(S2)
     => ( finite_finite_int(Ta)
       => ( ! [X2: int] :
              ( member_int(X2,Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S2)
               => ? [Xa: int] :
                    ( member_int(Xa,Ta)
                    & ( aa_int_complex(I2,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6013_sum__le__included,axiom,
    ! [S2: set_complex,Ta: set_complex,G: fun_complex_real,I2: fun_complex_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X2: complex] :
              ( member_complex(X2,Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_complex(I2,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S2)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6014_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_nat,G: fun_nat_rat,I2: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2)) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S2)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I2,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6015_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_int,G: fun_int_rat,I2: fun_int_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( finite_finite_int(Ta)
       => ( ! [X2: int] :
              ( member_int(X2,Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2)) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S2)
               => ? [Xa: int] :
                    ( member_int(Xa,Ta)
                    & ( aa_int_nat(I2,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6016_sum__le__included,axiom,
    ! [S2: set_nat,Ta: set_complex,G: fun_complex_rat,I2: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X2: complex] :
              ( member_complex(X2,Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2)) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_nat(I2,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S2)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6017_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_nat,G: fun_nat_rat,I2: fun_nat_int,F: fun_int_rat] :
      ( finite_finite_int(S2)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2)) )
         => ( ! [X2: int] :
                ( member_int(X2,S2)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_int(I2,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S2)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6018_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_int,G: fun_int_rat,I2: fun_int_int,F: fun_int_rat] :
      ( finite_finite_int(S2)
     => ( finite_finite_int(Ta)
       => ( ! [X2: int] :
              ( member_int(X2,Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X2)) )
         => ( ! [X2: int] :
                ( member_int(X2,S2)
               => ? [Xa: int] :
                    ( member_int(Xa,Ta)
                    & ( aa_int_int(I2,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S2)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6019_sum__le__included,axiom,
    ! [S2: set_int,Ta: set_complex,G: fun_complex_rat,I2: fun_complex_int,F: fun_int_rat] :
      ( finite_finite_int(S2)
     => ( finite3207457112153483333omplex(Ta)
       => ( ! [X2: complex] :
              ( member_complex(X2,Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X2)) )
         => ( ! [X2: int] :
                ( member_int(X2,S2)
               => ? [Xa: complex] :
                    ( member_complex(Xa,Ta)
                    & ( aa_complex_int(I2,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S2)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_6020_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( finite_finite_real(A2)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6021_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_real] :
      ( finite_finite_int(A2)
     => ( ! [X2: int] :
            ( member_int(X2,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X2)) )
       => ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
        <=> ! [X3: int] :
              ( member_int(X3,A2)
             => ( aa_int_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6022_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)) )
       => ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
        <=> ! [X3: complex] :
              ( member_complex(X3,A2)
             => ( aa_complex_real(F,X3) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6023_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
       => ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6024_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
       => ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A2)
             => ( aa_nat_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6025_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_rat] :
      ( finite_finite_int(A2)
     => ( ! [X2: int] :
            ( member_int(X2,A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X2)) )
       => ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: int] :
              ( member_int(X3,A2)
             => ( aa_int_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6026_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
       => ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
        <=> ! [X3: complex] :
              ( member_complex(X3,A2)
             => ( aa_complex_rat(F,X3) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6027_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2)) )
       => ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: real] :
              ( member_real(X3,A2)
             => ( aa_real_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6028_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( finite_finite_int(A2)
     => ( ! [X2: int] :
            ( member_int(X2,A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X2)) )
       => ( ( groups4541462559716669496nt_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: int] :
              ( member_int(X3,A2)
             => ( aa_int_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6029_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2)) )
       => ( ( groups5693394587270226106ex_nat(F,A2) = zero_zero_nat )
        <=> ! [X3: complex] :
              ( member_complex(X3,A2)
             => ( aa_complex_nat(F,X3) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_6030_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( finite_finite_int(A2)
     => ( ! [X2: int] :
            ( member_int(X2,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X2)),aa_int_real(G,X2)) )
       => ( ? [X: int] :
              ( member_int(X,A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6031_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X2)),aa_complex_real(G,X2)) )
       => ( ? [X: complex] :
              ( member_complex(X,A2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6032_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6033_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(A2)
     => ( ! [X2: int] :
            ( member_int(X2,A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2)) )
       => ( ? [X: int] :
              ( member_int(X,A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6034_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2)) )
       => ( ? [X: complex] :
              ( member_complex(X,A2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6035_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(A2)
     => ( ! [X2: int] :
            ( member_int(X2,A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2)) )
       => ( ? [X: int] :
              ( member_int(X,A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6036_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2)) )
       => ( ? [X: complex] :
              ( member_complex(X,A2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6037_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X2)),aa_nat_int(G,X2)) )
       => ( ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6038_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( finite3207457112153483333omplex(A2)
     => ( ! [X2: complex] :
            ( member_complex(X2,A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X2)),aa_complex_int(G,X2)) )
       => ( ? [X: complex] :
              ( member_complex(X,A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X)),aa_complex_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6039_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: fun_int_int,G: fun_int_int] :
      ( finite_finite_int(A2)
     => ( ! [X2: int] :
            ( member_int(X2,A2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X2)),aa_int_int(G,X2)) )
       => ( ? [X: int] :
              ( member_int(X,A2)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X)),aa_int_int(G,X)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_6040_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X23)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X15),Y15)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X2)),aa_nat_complex(G,X2)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_set_nat_complex(groups2073611262835488442omplex(H),S3)),aa_set_nat_complex(groups2073611262835488442omplex(G),S3)) ) ) ) ) ).

% sum.related
tff(fact_6041_sum_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S3: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,zero_zero_complex),zero_zero_complex)
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X15),X23)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y15),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X15),Y15)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,X23),Y23)) )
       => ( finite_finite_int(S3)
         => ( ! [X2: int] :
                ( member_int(X2,S3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X2)),aa_int_complex(G,X2)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups3049146728041665814omplex(H,S3)),groups3049146728041665814omplex(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6042_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( finite_finite_int(S3)
         => ( ! [X2: int] :
                ( member_int(X2,S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X2)),aa_int_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6043_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X2)),aa_complex_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6044_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X2)),aa_nat_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6045_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( finite_finite_int(S3)
         => ( ! [X2: int] :
                ( member_int(X2,S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X2)),aa_int_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups3906332499630173760nt_rat(H,S3)),groups3906332499630173760nt_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6046_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X15),Y15)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X2)),aa_complex_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6047_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( finite_finite_int(S3)
         => ( ! [X2: int] :
                ( member_int(X2,S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X2)),aa_int_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6048_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X15),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( finite3207457112153483333omplex(S3)
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X2)),aa_complex_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6049_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S3: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X15),X23)
              & aa_int_o(aa_int_fun_int_o(R,Y15),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X2)),aa_nat_int(G,X2)) )
           => aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S3)),groups3539618377306564664at_int(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_6050_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( member_complex(X2,A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X2)),aa_complex_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6051_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_real,G: fun_int_real] :
      ( finite_finite_int(A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( member_int(X2,A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X2)),aa_int_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6052_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X2)),aa_real_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6053_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( member_complex(X2,A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X2)),aa_complex_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6054_sum__strict__mono,axiom,
    ! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X2)),aa_nat_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6055_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( finite_finite_int(A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( member_int(X2,A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X2)),aa_int_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6056_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X2)),aa_real_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6057_sum__strict__mono,axiom,
    ! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( member_complex(X2,A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X2)),aa_complex_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6058_sum__strict__mono,axiom,
    ! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( finite_finite_int(A2)
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( member_int(X2,A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X2)),aa_int_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6059_sum__strict__mono,axiom,
    ! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(A2)
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( member_real(X2,A2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X2)),aa_real_nat(G,X2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ) ).

% sum_strict_mono
tff(fact_6060_ln__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_bound
tff(fact_6061_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_ge_zero
tff(fact_6062_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I2: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I2,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T2,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T2,T3))
                 => ( aa_real_real(J,aa_real_real(I2,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T2,T3))
                   => member_real(aa_real_real(I2,B2),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_complex(H,B2) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_real_complex(H,aa_real_real(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6063_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S3: set_real,I2: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( finite_finite_real(S4)
     => ( finite_finite_int(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_int_real(I2,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_int(aa_real_int(J,A3),minus_minus_set_int(T2,T3)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T2,T3))
                 => ( aa_real_int(J,aa_int_real(I2,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T2,T3))
                   => member_real(aa_int_real(I2,B2),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T3)
                       => ( aa_int_complex(H,B2) = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_int_complex(H,aa_real_int(J,A3)) = aa_real_complex(G,A3) ) )
                     => ( groups5754745047067104278omplex(G,S3) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6064_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S3: set_int,I2: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( finite_finite_int(S4)
     => ( finite_finite_real(T3)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_real_int(I2,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T2,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T2,T3))
                 => ( aa_int_real(J,aa_real_int(I2,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T2,T3))
                   => member_int(aa_real_int(I2,B2),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_complex(H,B2) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_real_complex(H,aa_int_real(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6065_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S3: set_int,I2: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( finite_finite_int(S4)
     => ( finite_finite_int(T3)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_int_int(I2,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_int(aa_int_int(J,A3),minus_minus_set_int(T2,T3)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T2,T3))
                 => ( aa_int_int(J,aa_int_int(I2,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T2,T3))
                   => member_int(aa_int_int(I2,B2),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_complex(G,A3) = zero_zero_complex ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T3)
                       => ( aa_int_complex(H,B2) = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_int_complex(H,aa_int_int(J,A3)) = aa_int_complex(G,A3) ) )
                     => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6066_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S3: set_real,I2: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_real_real(I2,aa_real_real(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_real(aa_real_real(J,A3),minus_minus_set_real(T2,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T2,T3))
                 => ( aa_real_real(J,aa_real_real(I2,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T2,T3))
                   => member_real(aa_real_real(I2,B2),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_real(H,B2) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_real_real(H,aa_real_real(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6067_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S3: set_real,I2: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_real,H: fun_int_real] :
      ( finite_finite_real(S4)
     => ( finite_finite_int(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_int_real(I2,aa_real_int(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_int(aa_real_int(J,A3),minus_minus_set_int(T2,T3)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T2,T3))
                 => ( aa_real_int(J,aa_int_real(I2,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T2,T3))
                   => member_real(aa_int_real(I2,B2),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T3)
                       => ( aa_int_real(H,B2) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_int_real(H,aa_real_int(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6068_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S3: set_real,I2: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( finite_finite_real(S4)
     => ( finite3207457112153483333omplex(T3)
       => ( ! [A3: real] :
              ( member_real(A3,minus_minus_set_real(S3,S4))
             => ( aa_complex_real(I2,aa_real_complex(J,A3)) = A3 ) )
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(S3,S4))
               => member_complex(aa_real_complex(J,A3),minus_811609699411566653omplex(T2,T3)) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(T2,T3))
                 => ( aa_real_complex(J,aa_complex_real(I2,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( member_complex(B2,minus_811609699411566653omplex(T2,T3))
                   => member_real(aa_complex_real(I2,B2),minus_minus_set_real(S3,S4)) )
               => ( ! [A3: real] :
                      ( member_real(A3,S4)
                     => ( aa_real_real(G,A3) = zero_zero_real ) )
                 => ( ! [B2: complex] :
                        ( member_complex(B2,T3)
                       => ( aa_complex_real(H,B2) = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( member_real(A3,S3)
                         => ( aa_complex_real(H,aa_real_complex(J,A3)) = aa_real_real(G,A3) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6069_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S3: set_int,I2: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_real,H: fun_real_real] :
      ( finite_finite_int(S4)
     => ( finite_finite_real(T3)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_real_int(I2,aa_int_real(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_real(aa_int_real(J,A3),minus_minus_set_real(T2,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T2,T3))
                 => ( aa_int_real(J,aa_real_int(I2,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T2,T3))
                   => member_int(aa_real_int(I2,B2),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_real(H,B2) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_real_real(H,aa_int_real(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6070_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S3: set_int,I2: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_real,H: fun_int_real] :
      ( finite_finite_int(S4)
     => ( finite_finite_int(T3)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_int_int(I2,aa_int_int(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_int(aa_int_int(J,A3),minus_minus_set_int(T2,T3)) )
           => ( ! [B2: int] :
                  ( member_int(B2,minus_minus_set_int(T2,T3))
                 => ( aa_int_int(J,aa_int_int(I2,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( member_int(B2,minus_minus_set_int(T2,T3))
                   => member_int(aa_int_int(I2,B2),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B2: int] :
                        ( member_int(B2,T3)
                       => ( aa_int_real(H,B2) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_int_real(H,aa_int_int(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6071_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_complex,S3: set_int,I2: fun_complex_int,J: fun_int_complex,T2: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( finite_finite_int(S4)
     => ( finite3207457112153483333omplex(T3)
       => ( ! [A3: int] :
              ( member_int(A3,minus_minus_set_int(S3,S4))
             => ( aa_complex_int(I2,aa_int_complex(J,A3)) = A3 ) )
         => ( ! [A3: int] :
                ( member_int(A3,minus_minus_set_int(S3,S4))
               => member_complex(aa_int_complex(J,A3),minus_811609699411566653omplex(T2,T3)) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(T2,T3))
                 => ( aa_int_complex(J,aa_complex_int(I2,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( member_complex(B2,minus_811609699411566653omplex(T2,T3))
                   => member_int(aa_complex_int(I2,B2),minus_minus_set_int(S3,S4)) )
               => ( ! [A3: int] :
                      ( member_int(A3,S4)
                     => ( aa_int_real(G,A3) = zero_zero_real ) )
                 => ( ! [B2: complex] :
                        ( member_complex(B2,T3)
                       => ( aa_complex_real(H,B2) = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( member_int(A3,S3)
                         => ( aa_complex_real(H,aa_int_complex(J,A3)) = aa_int_real(G,A3) ) )
                     => ( groups8778361861064173332t_real(G,S3) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_6072_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_6073_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_6074_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_6075_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_6076_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6077_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6078_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6079_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_6080_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_6081_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_6082_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_6083_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_6084_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6085_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6086_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6087_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_6088_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_6089_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_6090_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_6091_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_6092_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_6093_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_6094_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_6095_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_6096_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_6097_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_6098_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_6099_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_6100_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_6101_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_6102_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_6103_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_6104_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_6105_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_6106_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_6107_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_6108_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_le_numeral
tff(fact_6109_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,Mb)) ).

% neg_one_le_numeral
tff(fact_6110_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).

% neg_one_le_numeral
tff(fact_6111_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_le_numeral
tff(fact_6112_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_6113_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_6114_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_6115_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_6116_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_6117_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_6118_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_6119_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_6120_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6121_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6122_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6123_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_le_neg_numeral
tff(fact_6124_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_less_one
tff(fact_6125_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_less_one
tff(fact_6126_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_6127_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_6128_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_less_numeral
tff(fact_6129_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_less_numeral
tff(fact_6130_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).

% neg_one_less_numeral
tff(fact_6131_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,Mb)) ).

% neg_one_less_numeral
tff(fact_6132_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_6133_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_6134_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_6135_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_6136_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6137_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6138_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6139_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_one_less_neg_numeral
tff(fact_6140_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6141_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6142_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6143_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_6144_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_6145_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_6146_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_6147_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_6148_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_6149_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_6150_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_6151_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_6152_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_6153_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_6154_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_6155_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) )
      <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6156_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_6157_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_6158_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6159_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_6160_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_6161_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_6162_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_6163_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_6164_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_6165_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_6166_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_6167_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_6168_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_6169_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_6170_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_6171_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_6172_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(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_6173_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_6174_power__minus,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_minus
tff(fact_6175_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_minus
tff(fact_6176_power__minus,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_minus
tff(fact_6177_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_6178_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_minus
tff(fact_6179_power__minus__Bit0,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6180_power__minus__Bit0,axiom,
    ! [Xb: int,K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6181_power__minus__Bit0,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6182_power__minus__Bit0,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6183_power__minus__Bit0,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6184_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_real,B3: real,I2: real] :
      ( finite_finite_real(S2)
     => ( ! [I3: real] :
            ( member_real(I3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,S2) = B3 )
         => ( member_real(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6185_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_real,B3: real,I2: int] :
      ( finite_finite_int(S2)
     => ( ! [I3: int] :
            ( member_int(I3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,S2) = B3 )
         => ( member_int(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6186_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_real,B3: real,I2: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I3: complex] :
            ( member_complex(I3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,S2) = B3 )
         => ( member_complex(I2,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6187_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_rat,B3: rat,I2: real] :
      ( finite_finite_real(S2)
     => ( ! [I3: real] :
            ( member_real(I3,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,S2) = B3 )
         => ( member_real(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6188_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: fun_nat_rat,B3: rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,S2) = B3 )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6189_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_rat,B3: rat,I2: int] :
      ( finite_finite_int(S2)
     => ( ! [I3: int] :
            ( member_int(I3,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = B3 )
         => ( member_int(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6190_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_rat,B3: rat,I2: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I3: complex] :
            ( member_complex(I3,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,S2) = B3 )
         => ( member_complex(I2,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6191_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: fun_real_nat,B3: nat,I2: real] :
      ( finite_finite_real(S2)
     => ( ! [I3: real] :
            ( member_real(I3,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
       => ( ( groups1935376822645274424al_nat(F,S2) = B3 )
         => ( member_real(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6192_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: fun_int_nat,B3: nat,I2: int] :
      ( finite_finite_int(S2)
     => ( ! [I3: int] :
            ( member_int(I3,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = B3 )
         => ( member_int(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6193_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: fun_complex_nat,B3: nat,I2: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I3: complex] :
            ( member_complex(I3,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
       => ( ( groups5693394587270226106ex_nat(F,S2) = B3 )
         => ( member_complex(I2,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),B3) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_6194_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_real,I2: real] :
      ( finite_finite_real(S2)
     => ( ! [I3: real] :
            ( member_real(I3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
       => ( ( groups8097168146408367636l_real(F,S2) = zero_zero_real )
         => ( member_real(I2,S2)
           => ( aa_real_real(F,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6195_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_real,I2: int] :
      ( finite_finite_int(S2)
     => ( ! [I3: int] :
            ( member_int(I3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
       => ( ( groups8778361861064173332t_real(F,S2) = zero_zero_real )
         => ( member_int(I2,S2)
           => ( aa_int_real(F,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6196_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_real,I2: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I3: complex] :
            ( member_complex(I3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
       => ( ( groups5808333547571424918x_real(F,S2) = zero_zero_real )
         => ( member_complex(I2,S2)
           => ( aa_complex_real(F,I2) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6197_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_rat,I2: real] :
      ( finite_finite_real(S2)
     => ( ! [I3: real] :
            ( member_real(I3,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
       => ( ( groups1300246762558778688al_rat(F,S2) = zero_zero_rat )
         => ( member_real(I2,S2)
           => ( aa_real_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6198_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: fun_nat_rat,I2: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ! [I3: nat] :
            ( aa_set_nat_o(member_nat(I3),S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
       => ( ( groups2906978787729119204at_rat(F,S2) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I2),S2)
           => ( aa_nat_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6199_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_rat,I2: int] :
      ( finite_finite_int(S2)
     => ( ! [I3: int] :
            ( member_int(I3,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
       => ( ( groups3906332499630173760nt_rat(F,S2) = zero_zero_rat )
         => ( member_int(I2,S2)
           => ( aa_int_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6200_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_rat,I2: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I3: complex] :
            ( member_complex(I3,S2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
       => ( ( groups5058264527183730370ex_rat(F,S2) = zero_zero_rat )
         => ( member_complex(I2,S2)
           => ( aa_complex_rat(F,I2) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6201_sum__nonneg__0,axiom,
    ! [S2: set_real,F: fun_real_nat,I2: real] :
      ( finite_finite_real(S2)
     => ( ! [I3: real] :
            ( member_real(I3,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
       => ( ( groups1935376822645274424al_nat(F,S2) = zero_zero_nat )
         => ( member_real(I2,S2)
           => ( aa_real_nat(F,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6202_sum__nonneg__0,axiom,
    ! [S2: set_int,F: fun_int_nat,I2: int] :
      ( finite_finite_int(S2)
     => ( ! [I3: int] :
            ( member_int(I3,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
       => ( ( groups4541462559716669496nt_nat(F,S2) = zero_zero_nat )
         => ( member_int(I2,S2)
           => ( aa_int_nat(F,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6203_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: fun_complex_nat,I2: complex] :
      ( finite3207457112153483333omplex(S2)
     => ( ! [I3: complex] :
            ( member_complex(I3,S2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
       => ( ( groups5693394587270226106ex_nat(F,S2) = zero_zero_nat )
         => ( member_complex(I2,S2)
           => ( aa_complex_nat(F,I2) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_6204_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_complex] :
      ( finite_finite_int(A2)
     => ( groups3049146728041665814omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_jz(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6205_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_real] :
      ( finite_finite_int(A2)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ka(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6206_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kb(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6207_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_rat] :
      ( finite_finite_int(A2)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_kc(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6208_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kd(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6209_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: fun_int_nat] :
      ( finite_finite_int(A2)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ke(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6210_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kf(G)))) = groups5693394587270226106ex_nat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6211_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(A2)
     => ( groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_kg(G)))) = groups5690904116761175830ex_int(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6212_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(G),minus_minus_set_nat(A2,collect_nat(aTP_Lamm_kh(G)))) = aa_set_nat_complex(groups2073611262835488442omplex(G),A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6213_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,collect_nat(aTP_Lamm_ki(G)))) = groups2906978787729119204at_rat(G,A2) ) ) ).

% sum.setdiff_irrelevant
tff(fact_6214_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_6215_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_6216_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_6217_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_6218_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_6219_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% zmod_zminus1_eq_if
tff(fact_6220_sum__pos2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( member_real(I2,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2))
         => ( ! [I3: real] :
                ( member_real(I3,I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6221_sum__pos2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_real] :
      ( finite_finite_int(I6)
     => ( member_int(I2,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2))
         => ( ! [I3: int] :
                ( member_int(I3,I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6222_sum__pos2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I2,I6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2))
         => ( ! [I3: complex] :
                ( member_complex(I3,I6)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6223_sum__pos2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( member_real(I2,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2))
         => ( ! [I3: real] :
                ( member_real(I3,I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6224_sum__pos2,axiom,
    ! [I6: set_nat,I2: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( aa_set_nat_o(member_nat(I2),I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2))
         => ( ! [I3: nat] :
                ( aa_set_nat_o(member_nat(I3),I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6225_sum__pos2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_rat] :
      ( finite_finite_int(I6)
     => ( member_int(I2,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2))
         => ( ! [I3: int] :
                ( member_int(I3,I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6226_sum__pos2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I2,I6)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2))
         => ( ! [I3: complex] :
                ( member_complex(I3,I6)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6227_sum__pos2,axiom,
    ! [I6: set_real,I2: real,F: fun_real_nat] :
      ( finite_finite_real(I6)
     => ( member_real(I2,I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2))
         => ( ! [I3: real] :
                ( member_real(I3,I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6228_sum__pos2,axiom,
    ! [I6: set_int,I2: int,F: fun_int_nat] :
      ( finite_finite_int(I6)
     => ( member_int(I2,I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2))
         => ( ! [I3: int] :
                ( member_int(I3,I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6229_sum__pos2,axiom,
    ! [I6: set_complex,I2: complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(I6)
     => ( member_complex(I2,I6)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2))
         => ( ! [I3: complex] :
                ( member_complex(I3,I6)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6)) ) ) ) ) ).

% sum_pos2
tff(fact_6230_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( member_complex(I3,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6231_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_real] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( member_int(I3,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6232_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_real] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( member_real(I3,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6233_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( member_complex(I3,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6234_sum__pos,axiom,
    ! [I6: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I6)
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6235_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_rat] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( member_int(I3,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6236_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_rat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( member_real(I3,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6237_sum__pos,axiom,
    ! [I6: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(I6)
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( member_complex(I3,I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6238_sum__pos,axiom,
    ! [I6: set_int,F: fun_int_nat] :
      ( finite_finite_int(I6)
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( member_int(I3,I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6239_sum__pos,axiom,
    ! [I6: set_real,F: fun_real_nat] :
      ( finite_finite_real(I6)
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( member_real(I3,I6)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I6)) ) ) ) ).

% sum_pos
tff(fact_6240_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6241_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,T2) = groups5754745047067104278omplex(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6242_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6243_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6244_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6245_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_rat(G,X2) = aa_complex_rat(H,X2) ) )
           => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6246_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_nat(G,X2) = zero_zero_nat ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_nat(G,X2) = aa_real_nat(H,X2) ) )
           => ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6247_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_nat(G,X2) = aa_complex_nat(H,X2) ) )
           => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6248_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S3: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_int(G,X2) = zero_zero_int ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_int(G,X2) = aa_real_int(H,X2) ) )
           => ( groups1932886352136224148al_int(G,T2) = groups1932886352136224148al_int(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6249_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_int(G,X2) = aa_complex_int(H,X2) ) )
           => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6250_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => ( aa_nat_complex(G,X2) = aa_nat_complex(H,X2) ) )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(G),T2) = aa_set_nat_complex(groups2073611262835488442omplex(H),S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_6251_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_complex(G,X2) = aa_real_complex(H,X2) ) )
           => ( groups5754745047067104278omplex(G,S3) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6252_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6253_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6254_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,S3) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6255_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_rat(G,X2) = aa_complex_rat(H,X2) ) )
           => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6256_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_nat(H,X2) = zero_zero_nat ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_nat(G,X2) = aa_real_nat(H,X2) ) )
           => ( groups1935376822645274424al_nat(G,S3) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6257_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(H,X2) = zero_zero_nat ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_nat(G,X2) = aa_complex_nat(H,X2) ) )
           => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6258_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S3: set_real,H: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(T2)
     => ( ord_less_eq_set_real(S3,T2)
       => ( ! [X2: real] :
              ( member_real(X2,minus_minus_set_real(T2,S3))
             => ( aa_real_int(H,X2) = zero_zero_int ) )
         => ( ! [X2: real] :
                ( member_real(X2,S3)
               => ( aa_real_int(G,X2) = aa_real_int(H,X2) ) )
           => ( groups1932886352136224148al_int(G,S3) = groups1932886352136224148al_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6259_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S3: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(H,X2) = zero_zero_int ) )
         => ( ! [X2: complex] :
                ( member_complex(X2,S3)
               => ( aa_complex_int(G,X2) = aa_complex_int(H,X2) ) )
           => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6260_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_nat,S3: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(H,X2) = zero_zero_complex ) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => ( aa_nat_complex(G,X2) = aa_nat_complex(H,X2) ) )
           => ( aa_set_nat_complex(groups2073611262835488442omplex(G),S3) = aa_set_nat_complex(groups2073611262835488442omplex(H),T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_6261_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6262_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6263_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6264_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6265_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G),T2) = aa_set_nat_complex(groups2073611262835488442omplex(G),S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6266_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_rat(G,X2) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6267_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_int(G,X2) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T2) = groups3539618377306564664at_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6268_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( finite_finite_int(T2)
     => ( ord_less_eq_set_int(S3,T2)
       => ( ! [X2: int] :
              ( member_int(X2,minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6269_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( finite_finite_int(T2)
     => ( ord_less_eq_set_int(S3,T2)
       => ( ! [X2: int] :
              ( member_int(X2,minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6270_sum_Omono__neutral__right,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_rat] :
      ( finite_finite_int(T2)
     => ( ord_less_eq_set_int(S3,T2)
       => ( ! [X2: int] :
              ( member_int(X2,minus_minus_set_int(T2,S3))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_6271_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_real] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6272_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6273_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6274_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S3: set_complex,G: fun_complex_int] :
      ( finite3207457112153483333omplex(T2)
     => ( ord_le211207098394363844omplex(S3,T2)
       => ( ! [X2: complex] :
              ( member_complex(X2,minus_811609699411566653omplex(T2,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6275_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_complex(G,X2) = zero_zero_complex ) )
         => ( aa_set_nat_complex(groups2073611262835488442omplex(G),S3) = aa_set_nat_complex(groups2073611262835488442omplex(G),T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6276_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_rat(G,X2) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S3) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6277_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(S3),T2)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),minus_minus_set_nat(T2,S3))
             => ( aa_nat_int(G,X2) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S3) = groups3539618377306564664at_int(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6278_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_complex] :
      ( finite_finite_int(T2)
     => ( ord_less_eq_set_int(S3,T2)
       => ( ! [X2: int] :
              ( member_int(X2,minus_minus_set_int(T2,S3))
             => ( aa_int_complex(G,X2) = zero_zero_complex ) )
         => ( groups3049146728041665814omplex(G,S3) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6279_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_real] :
      ( finite_finite_int(T2)
     => ( ord_less_eq_set_int(S3,T2)
       => ( ! [X2: int] :
              ( member_int(X2,minus_minus_set_int(T2,S3))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6280_sum_Omono__neutral__left,axiom,
    ! [T2: set_int,S3: set_int,G: fun_int_rat] :
      ( finite_finite_int(T2)
     => ( ord_less_eq_set_int(S3,T2)
       => ( ! [X2: int] :
              ( member_int(X2,minus_minus_set_int(T2,S3))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(G,T2) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_6281_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_complex(H,B2) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,C2) = groups5754745047067104278omplex(H,C2) )
               => ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6282_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_real(H,B2) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) )
               => ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6283_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_real(H,B2) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) )
               => ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6284_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) )
               => ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6285_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) )
               => ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6286_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) )
               => ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6287_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) )
               => ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6288_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_int(H,B2) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) )
               => ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6289_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_int(H,B2) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) )
               => ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6290_sum_Osame__carrierI,axiom,
    ! [C2: set_nat,A2: set_nat,B3: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C2)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),minus_minus_set_nat(C2,A2))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(C2,B3))
                 => ( aa_nat_complex(H,B2) = zero_zero_complex ) )
             => ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),C2) = aa_set_nat_complex(groups2073611262835488442omplex(H),C2) )
               => ( aa_set_nat_complex(groups2073611262835488442omplex(G),A2) = aa_set_nat_complex(groups2073611262835488442omplex(H),B3) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_6291_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_complex(H,B2) = zero_zero_complex ) )
             => ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B3) )
              <=> ( groups5754745047067104278omplex(G,C2) = groups5754745047067104278omplex(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6292_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_real(H,B2) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B3) )
              <=> ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6293_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_real(G,A3) = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_real(H,B2) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B3) )
              <=> ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6294_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B3) )
              <=> ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6295_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_rat(G,A3) = zero_zero_rat ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B3) )
              <=> ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6296_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B3) )
              <=> ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6297_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_nat(G,A3) = zero_zero_nat ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A2) = groups5693394587270226106ex_nat(H,B3) )
              <=> ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6298_sum_Osame__carrier,axiom,
    ! [C2: set_real,A2: set_real,B3: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A2,C2)
       => ( ord_less_eq_set_real(B3,C2)
         => ( ! [A3: real] :
                ( member_real(A3,minus_minus_set_real(C2,A2))
               => ( aa_real_int(G,A3) = zero_zero_int ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B3))
                 => ( aa_real_int(H,B2) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A2) = groups1932886352136224148al_int(H,B3) )
              <=> ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6299_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A2: set_complex,B3: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( finite3207457112153483333omplex(C2)
     => ( ord_le211207098394363844omplex(A2,C2)
       => ( ord_le211207098394363844omplex(B3,C2)
         => ( ! [A3: complex] :
                ( member_complex(A3,minus_811609699411566653omplex(C2,A2))
               => ( aa_complex_int(G,A3) = zero_zero_int ) )
           => ( ! [B2: complex] :
                  ( member_complex(B2,minus_811609699411566653omplex(C2,B3))
                 => ( aa_complex_int(H,B2) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A2) = groups5690904116761175830ex_int(H,B3) )
              <=> ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6300_sum_Osame__carrier,axiom,
    ! [C2: set_nat,A2: set_nat,B3: set_nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),C2)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B3),C2)
         => ( ! [A3: nat] :
                ( aa_set_nat_o(member_nat(A3),minus_minus_set_nat(C2,A2))
               => ( aa_nat_complex(G,A3) = zero_zero_complex ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(C2,B3))
                 => ( aa_nat_complex(H,B2) = zero_zero_complex ) )
             => ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),A2) = aa_set_nat_complex(groups2073611262835488442omplex(H),B3) )
              <=> ( aa_set_nat_complex(groups2073611262835488442omplex(G),C2) = aa_set_nat_complex(groups2073611262835488442omplex(H),C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_6301_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_real] :
      ( ord_le211207098394363844omplex(B3,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B3))),groups5808333547571424918x_real(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6302_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_rat] :
      ( ord_le211207098394363844omplex(B3,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5058264527183730370ex_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B3))),groups5058264527183730370ex_rat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6303_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_nat] :
      ( ord_le211207098394363844omplex(B3,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B3))),groups5693394587270226106ex_nat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6304_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: fun_complex_int] :
      ( ord_le211207098394363844omplex(B3,A2)
     => ( finite3207457112153483333omplex(A2)
       => ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B3))),groups5690904116761175830ex_int(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6305_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B3),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups2906978787729119204at_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B3))),groups2906978787729119204at_rat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6306_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B3),A2)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( groups3539618377306564664at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B3))),groups3539618377306564664at_int(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6307_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: fun_int_real] :
      ( ord_less_eq_set_int(B3,A2)
     => ( 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,B3))),groups8778361861064173332t_real(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6308_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: fun_int_rat] :
      ( ord_less_eq_set_int(B3,A2)
     => ( finite_finite_int(A2)
       => ( groups3906332499630173760nt_rat(G,A2) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B3))),groups3906332499630173760nt_rat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6309_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: fun_int_nat] :
      ( ord_less_eq_set_int(B3,A2)
     => ( 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,B3))),groups4541462559716669496nt_nat(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6310_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: fun_int_int] :
      ( ord_less_eq_set_int(B3,A2)
     => ( 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,B3))),groups4538972089207619220nt_int(G,B3)) ) ) ) ).

% sum.subset_diff
tff(fact_6311_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(A2)
     => ( ord_le211207098394363844omplex(B3,A2)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A2,B3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6312_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(A2)
     => ( ord_le211207098394363844omplex(B3,A2)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,B3)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6313_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(A2)
     => ( ord_le211207098394363844omplex(B3,A2)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6314_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B3),A2)
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B3)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6315_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B3),A2)
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6316_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_real] :
      ( finite_finite_int(A2)
     => ( ord_less_eq_set_int(B3,A2)
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A2,B3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6317_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_rat] :
      ( finite_finite_int(A2)
     => ( ord_less_eq_set_int(B3,A2)
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B3)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6318_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: fun_int_int] :
      ( finite_finite_int(A2)
     => ( ord_less_eq_set_int(B3,A2)
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B3)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6319_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: fun_complex_complex] :
      ( finite3207457112153483333omplex(A2)
     => ( ord_le211207098394363844omplex(B3,A2)
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,B3)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(F,B3)) ) ) ) ).

% sum_diff
tff(fact_6320_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B3),A2)
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A2,B3)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2)),aa_set_nat_real(groups6591440286371151544t_real(F),B3)) ) ) ) ).

% sum_diff
tff(fact_6321_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_6322_ln__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_6323_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6324_ln__div,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_6325_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6326_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6327_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6328_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6329_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6330_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6331_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6332_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6333_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6334_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_6335_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_6336_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_6337_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6338_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6339_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6340_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6341_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6342_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6343_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xb: 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,Xb),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,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6344_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z))),Y) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6345_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: 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(Xb),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6346_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6347_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),Z))),B) = $ite(Z = zero_zero_complex,B,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6348_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6349_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6350_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),Z))),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6351_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6352_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6353_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z: complex,B: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),Z)),B) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6354_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6355_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: 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,Xb),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,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6356_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z))),Y) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6357_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: 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(Xb),Z))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6358_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_6359_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_6360_power2__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6361_power2__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6362_power2__eq__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6363_power2__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),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))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6364_power2__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),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))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6365_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_6366_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_real] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B3,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6367_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( ! [B2: complex] :
              ( member_complex(B2,minus_811609699411566653omplex(B3,A2))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6368_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_rat] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6369_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( ! [B2: complex] :
              ( member_complex(B2,minus_811609699411566653omplex(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6370_sum__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B3,A2))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6371_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_nat] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B3,A2))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6372_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( ! [B2: complex] :
              ( member_complex(B2,minus_811609699411566653omplex(B3,A2))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6373_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: fun_real_int] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B3,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6374_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( ! [B2: complex] :
              ( member_complex(B2,minus_811609699411566653omplex(B3,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6375_sum__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B3,A2))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B3)) ) ) ) ).

% sum_mono2
tff(fact_6376_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6377_ln__diff__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),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,Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_6378_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6379_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6380_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6381_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6382_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6383_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6384_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6385_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6386_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6387_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_6388_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_6389_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_6390_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6391_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6392_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6393_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6394_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_6395_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_6396_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_6397_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_6398_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6399_uminus__power__if,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ).

% uminus_power_if
tff(fact_6400_uminus__power__if,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ).

% uminus_power_if
tff(fact_6401_uminus__power__if,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb))) ).

% uminus_power_if
tff(fact_6402_uminus__power__if,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb))) ).

% uminus_power_if
tff(fact_6403_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).

% uminus_power_if
tff(fact_6404_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6405_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6406_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6407_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6408_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_6409_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_6410_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6411_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_6412_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_6413_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_6414_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_6415_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_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_6416_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: fun_real_real] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( member_real(B,minus_minus_set_real(B3,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,B))
           => ( ! [X2: real] :
                  ( member_real(X2,B3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6417_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( member_complex(B,minus_811609699411566653omplex(B3,A2))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,B))
           => ( ! [X2: complex] :
                  ( member_complex(X2,B3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6418_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: fun_real_rat] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( member_real(B,minus_minus_set_real(B3,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,B))
           => ( ! [X2: real] :
                  ( member_real(X2,B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6419_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( member_complex(B,minus_811609699411566653omplex(B3,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B))
           => ( ! [X2: complex] :
                  ( member_complex(X2,B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6420_sum__strict__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,B: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B3,A2))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,B))
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(member_nat(X2),B3)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6421_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: fun_real_nat] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( member_real(B,minus_minus_set_real(B3,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,B))
           => ( ! [X2: real] :
                  ( member_real(X2,B3)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X2)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6422_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( member_complex(B,minus_811609699411566653omplex(B3,A2))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B))
           => ( ! [X2: complex] :
                  ( member_complex(X2,B3)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X2)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6423_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: fun_real_int] :
      ( finite_finite_real(B3)
     => ( ord_less_eq_set_real(A2,B3)
       => ( member_real(B,minus_minus_set_real(B3,A2))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,B))
           => ( ! [X2: real] :
                  ( member_real(X2,B3)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X2)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6424_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: fun_complex_int] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(A2,B3)
       => ( member_complex(B,minus_811609699411566653omplex(B3,A2))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_complex_int(F,B))
           => ( ! [X2: complex] :
                  ( member_complex(X2,B3)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X2)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6425_sum__strict__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,B: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
       => ( aa_set_nat_o(member_nat(B),minus_minus_set_nat(B3,A2))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,B))
           => ( ! [X2: nat] :
                  ( aa_set_nat_o(member_nat(X2),B3)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X2)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B3)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_6426_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_6427_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_6428_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int))),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_6429_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6430_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_6431_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6432_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_6433_square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_6434_square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_6435_square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_6436_square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_6437_minus__power__mult__self,axiom,
    ! [A: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6438_minus__power__mult__self,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6439_minus__power__mult__self,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6440_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6441_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_6442_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_6443_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_6444_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_6445_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_6446_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_6447_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_6448_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_6449_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_6450_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_6451_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_6452_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_6453_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_6454_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_6455_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_6456_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_6457_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_6458_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_6459_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_6460_crossproduct__eq,axiom,
    ! [W: real,Y: real,Xb: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_6461_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,Xb: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Xb),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Z)),aa_rat_rat(times_times_rat(Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_6462_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,Xb: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_6463_crossproduct__eq,axiom,
    ! [W: int,Y: int,Xb: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_6464_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_6465_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_6466_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_6467_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_6468_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_6469_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_6470_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ey(L)),Qr) ).

% divmod_step_nat_def
tff(fact_6471_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6472_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_6473_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ez(L)),Qr) ).

% divmod_step_int_def
tff(fact_6474_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_6475_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6476_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6477_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_kj),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6478_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_kk),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6479_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_kl,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6480_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_km),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_6481_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,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_ri631733984087533419it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_6482_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),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_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6483_semiring__norm_I90_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit1,Mb) = aa_num_num(bit1,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(90)
tff(fact_6484_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_6485_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_6486_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_6487_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_6488_case__prodI,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B))
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B))) ) ).

% case_prodI
tff(fact_6489_case__prodI,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(F,A),B)
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B)) ) ).

% case_prodI
tff(fact_6490_case__prodI,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(F,A),B)
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B)) ) ).

% case_prodI
tff(fact_6491_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) ) ).

% case_prodI
tff(fact_6492_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_6493_case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,C: fun_Co480733329897759773un_o_o] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B2)) )
         => aa_o_o(aa_Cod6379977994291289310un_o_o(C,A3),(B2)) )
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2) ) ).

% case_prodI2
tff(fact_6494_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = product_Pair_num_num(A3,B2) )
         => aa_num_o(aa_num_fun_num_o(C,A3),B2) )
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).

% case_prodI2
tff(fact_6495_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A3: nat,B2: num] :
          ( ( P2 = product_Pair_nat_num(A3,B2) )
         => aa_num_o(aa_nat_fun_num_o(C,A3),B2) )
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).

% case_prodI2
tff(fact_6496_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A3: nat,B2: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A3),B2) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_6497_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A3: int,B2: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B2) )
         => aa_int_o(aa_int_fun_int_o(C,A3),B2) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_6498_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,A: code_integer,B: $o] :
      ( member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A),(B)))
     => member_complex(Z,produc1043322548047392435omplex(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6499_mem__case__prodI,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,A: code_integer,B: $o] :
      ( member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A),(B)))
     => member_real(Z,produc242741666403216561t_real(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6500_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,A: code_integer,B: $o] :
      ( aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A),(B)))
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6501_mem__case__prodI,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,A: code_integer,B: $o] :
      ( member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A),(B)))
     => member_int(Z,produc1253318751659547953et_int(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6502_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,A: num,B: num] :
      ( member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,A),B))
     => member_complex(Z,produc2866383454006189126omplex(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6503_mem__case__prodI,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,A: num,B: num] :
      ( member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,A),B))
     => member_real(Z,produc8296048397933160132t_real(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6504_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,A: num,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6505_mem__case__prodI,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,A: num,B: num] :
      ( member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,A),B))
     => member_int(Z,produc6406642877701697732et_int(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6506_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,A: nat,B: num] :
      ( member_complex(Z,aa_num_set_complex(aa_nat4347971807543021193omplex(C,A),B))
     => member_complex(Z,produc6231982587499038204omplex(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6507_mem__case__prodI,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,A: nat,B: num] :
      ( member_real(Z,aa_num_set_real(aa_nat1437378620602899975t_real(C,A),B))
     => member_real(Z,produc1435849484188172666t_real(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6508_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: complex,C: fun_Co5015990272074005067omplex] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B2)) )
         => member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A3),(B2))) )
     => member_complex(Z,produc1043322548047392435omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6509_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: real,C: fun_Co5368275265270938953t_real] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B2)) )
         => member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A3),(B2))) )
     => member_real(Z,produc242741666403216561t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6510_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: nat,C: fun_Co7401227239307276909et_nat] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B2)) )
         => aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A3),(B2))) )
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6511_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: int,C: fun_Co2587678820519358153et_int] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A3,(B2)) )
         => member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A3),(B2))) )
     => member_int(Z,produc1253318751659547953et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6512_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: complex,C: fun_nu9041852840018610706omplex] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = product_Pair_num_num(A3,B2) )
         => member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,A3),B2)) )
     => member_complex(Z,produc2866383454006189126omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6513_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: fun_nu4698968662094489488t_real] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = product_Pair_num_num(A3,B2) )
         => member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,A3),B2)) )
     => member_real(Z,produc8296048397933160132t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6514_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: fun_nu530747996811530292et_nat] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = product_Pair_num_num(A3,B2) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A3),B2)) )
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6515_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: fun_nu4940571614878387344et_int] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = product_Pair_num_num(A3,B2) )
         => member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,A3),B2)) )
     => member_int(Z,produc6406642877701697732et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6516_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: complex,C: fun_na3925014923085203400omplex] :
      ( ! [A3: nat,B2: num] :
          ( ( P2 = product_Pair_nat_num(A3,B2) )
         => member_complex(Z,aa_num_set_complex(aa_nat4347971807543021193omplex(C,A3),B2)) )
     => member_complex(Z,produc6231982587499038204omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6517_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: real,C: fun_na3666575854997362246t_real] :
      ( ! [A3: nat,B2: num] :
          ( ( P2 = product_Pair_nat_num(A3,B2) )
         => member_real(Z,aa_num_set_real(aa_nat1437378620602899975t_real(C,A3),B2)) )
     => member_real(Z,produc1435849484188172666t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6518_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,Xb: product_prod_nat_nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B2) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A3),B2),Xb) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Xb) ) ).

% case_prodI2'
tff(fact_6519_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_6520_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_6521_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_6522_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_6523_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_6524_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit1,Mb) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_6525_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit0,Mb) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_6526_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa_num_num(bit1,Mb) != one ).

% semiring_norm(86)
tff(fact_6527_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_6528_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Nb)) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% abs_numeral
tff(fact_6529_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_numeral
tff(fact_6530_abs__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_numeral
tff(fact_6531_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_6532_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_6533_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_6534_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_6535_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_6536_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_6537_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_6538_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_6539_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_6540_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_6541_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_6542_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_6543_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_6544_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_6545_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_6546_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_6547_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_6548_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_6549_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_6550_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_6551_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_6552_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_6553_abs__dvd__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,Mb)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6554_abs__dvd__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,Mb)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6555_abs__dvd__iff,axiom,
    ! [Mb: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,Mb)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6556_abs__dvd__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(Mb)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6557_dvd__abs__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6558_dvd__abs__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6559_dvd__abs__iff,axiom,
    ! [Mb: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),aa_rat_rat(abs_abs_rat,K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6560_dvd__abs__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),abs_abs_Code_integer(K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6561_abs__bool__eq,axiom,
    ! [P: $o] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real((P))) = zero_n3304061248610475627l_real((P)) ).

% abs_bool_eq
tff(fact_6562_abs__bool__eq,axiom,
    ! [P: $o] : aa_rat_rat(abs_abs_rat,zero_n2052037380579107095ol_rat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% abs_bool_eq
tff(fact_6563_abs__bool__eq,axiom,
    ! [P: $o] : aa_int_int(abs_abs_int,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% abs_bool_eq
tff(fact_6564_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_Code_integer(zero_n356916108424825756nteger((P))) = zero_n356916108424825756nteger((P)) ).

% abs_bool_eq
tff(fact_6565_tanh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% tanh_real_le_iff
tff(fact_6566_semiring__norm_I80_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(80)
tff(fact_6567_semiring__norm_I73_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(73)
tff(fact_6568_abs__sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(abs_abs_int,groups4538972089207619220nt_int(aTP_Lamm_kn(F),A2)) = groups4538972089207619220nt_int(aTP_Lamm_kn(F),A2) ).

% abs_sum_abs
tff(fact_6569_abs__sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ko(F)),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ko(F)),A2) ).

% abs_sum_abs
tff(fact_6570_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_6571_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_6572_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_6573_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_6574_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_6575_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_6576_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_6577_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_6578_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_6579_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6580_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6581_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6582_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_6583_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_6584_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_6585_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_6586_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_neg_numeral
tff(fact_6587_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_6588_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_neg_numeral
tff(fact_6589_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Nb))) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% abs_neg_numeral
tff(fact_6590_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_6591_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_6592_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_6593_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6594_abs__power__minus,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% abs_power_minus
tff(fact_6595_abs__power__minus,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% abs_power_minus
tff(fact_6596_abs__power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% abs_power_minus
tff(fact_6597_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% abs_power_minus
tff(fact_6598_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(7)
tff(fact_6599_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)) ).

% semiring_norm(9)
tff(fact_6600_semiring__norm_I14_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_6601_semiring__norm_I15_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),Nb)) ).

% semiring_norm(15)
tff(fact_6602_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_6603_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_6604_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_6605_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).

% semiring_norm(70)
tff(fact_6606_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_6607_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_6608_sum__abs,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_kn(F),A2)) ).

% sum_abs
tff(fact_6609_sum__abs,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ko(F)),A2)) ).

% sum_abs
tff(fact_6610_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6611_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6612_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6613_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6614_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_6615_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_6616_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_6617_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_6618_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_6619_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)),one)) ).

% semiring_norm(10)
tff(fact_6620_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),one)) ).

% semiring_norm(8)
tff(fact_6621_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).

% semiring_norm(5)
tff(fact_6622_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,Nb),one)) ).

% semiring_norm(4)
tff(fact_6623_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_6624_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_kn(F),A2)) ).

% sum_abs_ge_zero
tff(fact_6625_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ko(F)),A2)) ).

% sum_abs_ge_zero
tff(fact_6626_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,Mb),Nb)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_6627_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_6628_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_6629_numeral__div__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(Mb,Nb))) ).

% numeral_div_minus_numeral
tff(fact_6630_minus__numeral__div__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_div_numeral
tff(fact_6631_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6632_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6633_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6634_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6635_abs__power2,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6636_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6637_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6638_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6639_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6640_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6641_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6642_power2__abs,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6643_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_set_nat_complex(groups2073611262835488442omplex(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_6644_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_6645_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_6646_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_6647_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_6648_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6649_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6650_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6651_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Mb)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_6652_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_6653_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique3479559517661332726nteger(Mb,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_6654_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A2: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_kp(C)),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_6655_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_kq(C),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_6656_sum__zero__power,axiom,
    ! [C: fun_nat_real,A2: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kr(C)),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_6657_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6658_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6659_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6660_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6661_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_6662_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_6663_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_6664_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_6665_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_6666_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_6667_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_6668_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_6669_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_6670_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_6671_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A2: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ks(C),D)),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
        zero_zero_complex ) ).

% sum_zero_power'
tff(fact_6672_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A2: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kt(C),D),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_6673_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A2: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ku(C),D)),A2) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A2)
        & aa_set_nat_o(member_nat(zero_zero_nat),A2) ),
        aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_6674_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_6675_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_6676_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_6677_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6678_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6679_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6680_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6681_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6682_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6683_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Nb)),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_ri631733984087533419it_int,Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_6684_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_kv),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6685_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_kw),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6686_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_kx,unique3479559517661332726nteger(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6687_Collect__neg__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aTP_Lamm_ky(P)) = uminus6221592323253981072nt_int(collec213857154873943460nt_int(P)) ).

% Collect_neg_eq
tff(fact_6688_Collect__neg__eq,axiom,
    ! [P: fun_complex_o] : collect_complex(aTP_Lamm_kz(P)) = uminus8566677241136511917omplex(collect_complex(P)) ).

% Collect_neg_eq
tff(fact_6689_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_o] : collect_set_nat(aTP_Lamm_la(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_6690_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_lb(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_6691_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : collect_int(aTP_Lamm_lc(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_6692_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_ld(A2)) ).

% Compl_eq
tff(fact_6693_Compl__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A2) = collec213857154873943460nt_int(aTP_Lamm_le(A2)) ).

% Compl_eq
tff(fact_6694_Compl__eq,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(aTP_Lamm_lf(A2)) ).

% Compl_eq
tff(fact_6695_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_lg(A2)) ).

% Compl_eq
tff(fact_6696_Compl__eq,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(aTP_Lamm_lh(A2)) ).

% Compl_eq
tff(fact_6697_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_li(A2)) ).

% Compl_eq
tff(fact_6698_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_a(A2))) ).

% uminus_set_def
tff(fact_6699_uminus__set__def,axiom,
    ! [A2: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A2) = collec213857154873943460nt_int(uminus103585356917985579_int_o(aTP_Lamm_aa(A2))) ).

% uminus_set_def
tff(fact_6700_uminus__set__def,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_6701_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_6702_uminus__set__def,axiom,
    ! [A2: set_nat] : uminus5710092332889474511et_nat(A2) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_6703_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_6704_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_6705_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_6706_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_6707_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_6708_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_6709_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_6710_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_6711_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_6712_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_6713_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_6714_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_6715_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_6716_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_6717_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_6718_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_6719_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_6720_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_6721_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_6722_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_6723_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_6724_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_6725_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_6726_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_6727_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_6728_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_6729_power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) ).

% power_abs
tff(fact_6730_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).

% power_abs
tff(fact_6731_power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) ).

% power_abs
tff(fact_6732_power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb) ).

% power_abs
tff(fact_6733_abs__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6734_abs__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,Xb) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6735_abs__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Xb) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6736_abs__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(Xb) = abs_abs_Code_integer(Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6737_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6738_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6739_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6740_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6741_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,P2: produc6271795597528267376eger_o] :
      ( member_complex(Z,produc1043322548047392435omplex(C,P2))
     => ~ ! [X2: code_integer,Y4: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y4)) )
           => ~ member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,X2),(Y4))) ) ) ).

% mem_case_prodE
tff(fact_6742_mem__case__prodE,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,P2: produc6271795597528267376eger_o] :
      ( member_real(Z,produc242741666403216561t_real(C,P2))
     => ~ ! [X2: code_integer,Y4: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y4)) )
           => ~ member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,X2),(Y4))) ) ) ).

% mem_case_prodE
tff(fact_6743_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,P2: produc6271795597528267376eger_o] :
      ( aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2))
     => ~ ! [X2: code_integer,Y4: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y4)) )
           => ~ aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,X2),(Y4))) ) ) ).

% mem_case_prodE
tff(fact_6744_mem__case__prodE,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,P2: produc6271795597528267376eger_o] :
      ( member_int(Z,produc1253318751659547953et_int(C,P2))
     => ~ ! [X2: code_integer,Y4: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y4)) )
           => ~ member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,X2),(Y4))) ) ) ).

% mem_case_prodE
tff(fact_6745_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,P2: product_prod_num_num] :
      ( member_complex(Z,produc2866383454006189126omplex(C,P2))
     => ~ ! [X2: num,Y4: num] :
            ( ( P2 = product_Pair_num_num(X2,Y4) )
           => ~ member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_6746_mem__case__prodE,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
      ( member_real(Z,produc8296048397933160132t_real(C,P2))
     => ~ ! [X2: num,Y4: num] :
            ( ( P2 = product_Pair_num_num(X2,Y4) )
           => ~ member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_6747_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
      ( aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2))
     => ~ ! [X2: num,Y4: num] :
            ( ( P2 = product_Pair_num_num(X2,Y4) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_6748_mem__case__prodE,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
      ( member_int(Z,produc6406642877701697732et_int(C,P2))
     => ~ ! [X2: num,Y4: num] :
            ( ( P2 = product_Pair_num_num(X2,Y4) )
           => ~ member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_6749_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,P2: product_prod_nat_num] :
      ( member_complex(Z,produc6231982587499038204omplex(C,P2))
     => ~ ! [X2: nat,Y4: num] :
            ( ( P2 = product_Pair_nat_num(X2,Y4) )
           => ~ member_complex(Z,aa_num_set_complex(aa_nat4347971807543021193omplex(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_6750_mem__case__prodE,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,P2: product_prod_nat_num] :
      ( member_real(Z,produc1435849484188172666t_real(C,P2))
     => ~ ! [X2: nat,Y4: num] :
            ( ( P2 = product_Pair_nat_num(X2,Y4) )
           => ~ member_real(Z,aa_num_set_real(aa_nat1437378620602899975t_real(C,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_6751_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_6752_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_6753_case__prodE,axiom,
    ! [C: fun_Co480733329897759773un_o_o,P2: produc6271795597528267376eger_o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2)
     => ~ ! [X2: code_integer,Y4: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X2,(Y4)) )
           => ~ aa_o_o(aa_Cod6379977994291289310un_o_o(C,X2),(Y4)) ) ) ).

% case_prodE
tff(fact_6754_case__prodE,axiom,
    ! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
     => ~ ! [X2: num,Y4: num] :
            ( ( P2 = product_Pair_num_num(X2,Y4) )
           => ~ aa_num_o(aa_num_fun_num_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_6755_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
     => ~ ! [X2: nat,Y4: num] :
            ( ( P2 = product_Pair_nat_num(X2,Y4) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_6756_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X2: nat,Y4: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_6757_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X2: int,Y4: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X2),Y4) ) ) ).

% case_prodE
tff(fact_6758_case__prodD,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B)))
     => aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B)) ) ).

% case_prodD
tff(fact_6759_case__prodD,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B))
     => aa_num_o(aa_num_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6760_case__prodD,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B))
     => aa_num_o(aa_nat_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6761_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_6762_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_6763_case__prodD_H,axiom,
    ! [R: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R,A),B),C) ) ).

% case_prodD'
tff(fact_6764_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Z)
     => ~ ! [X2: nat,Y4: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X2),Y4),Z) ) ) ).

% case_prodE'
tff(fact_6765_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_6766_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_6767_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_6768_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_6769_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_6770_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_6771_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_6772_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_6773_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_6774_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_6775_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_6776_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_6777_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_6778_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_6779_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_6780_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_6781_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_6782_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_6783_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_6784_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_6785_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_6786_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6787_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6788_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6789_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_6790_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq3
tff(fact_6791_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq3
tff(fact_6792_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq3
tff(fact_6793_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_6794_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% abs_triangle_ineq2
tff(fact_6795_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq2
tff(fact_6796_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq2
tff(fact_6797_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_6798_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_6799_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_6800_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_6801_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_6802_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_6803_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_6804_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_6805_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_6806_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_6807_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_6808_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_6809_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_6810_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_6811_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_6812_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_6813_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_6814_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_6815_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_6816_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_6817_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_6818_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_6819_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A2)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X2)) = aa_nat_nat(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_6820_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A2)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A2)
           => ( aa_nat_real(F,aa_nat_nat(suc,X2)) = aa_nat_real(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A2) = aa_set_nat_real(groups6591440286371151544t_real(G),A2) ) ) ) ).

% sum_cong_Suc
tff(fact_6821_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_6822_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),one)
           => ( ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit0,N))
             => ( ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M),aa_num_num(bit1,N))
               => ( ! [M: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),one)
                 => ( ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit0,N))
                   => ~ ! [M: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_6823_sin__bound__lemma,axiom,
    ! [Xb: real,Y: real,U: real,V: real] :
      ( ( Xb = Y )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_6824_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X2: complex] :
          ( member_complex(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X2)),aa_complex_nat(F,X2)) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_lj(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ).

% sum_subtractf_nat
tff(fact_6825_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X2: real] :
          ( member_real(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X2)),aa_real_nat(F,X2)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_lk(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_6826_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X2: set_nat] :
          ( member_set_nat(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X2)),aa_set_nat_nat(F,X2)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ll(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_6827_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X2)),aa_int_nat(F,X2)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lm(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_6828_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X2)),aa_nat_nat(F,X2)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ln(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_6829_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lo(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6830_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_6831_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lq(G),K)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6832_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lr(G),K)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_6833_dense__eq0__I,axiom,
    ! [Xb: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),E2) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_6834_dense__eq0__I,axiom,
    ! [Xb: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),E2) )
     => ( Xb = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6835_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_6836_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_6837_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_6838_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_6839_abs__mult__pos,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),Xb) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6840_abs__mult__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),Xb) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6841_abs__mult__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y)),Xb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6842_abs__mult__pos,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6843_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_6844_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6845_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6846_zero__le__power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6847_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_6848_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6849_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6850_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_6851_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6852_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6853_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6854_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6855_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6856_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6857_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6858_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6859_abs__div__pos,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Xb)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_6860_abs__div__pos,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xb)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_6861_abs__if__raw,axiom,
    ! [X: real] :
      aa_real_real(abs_abs_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real),aa_real_real(uminus_uminus_real,X),X) ).

% abs_if_raw
tff(fact_6862_abs__if__raw,axiom,
    ! [X: int] :
      aa_int_int(abs_abs_int,X) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int),aa_int_int(uminus_uminus_int,X),X) ).

% abs_if_raw
tff(fact_6863_abs__if__raw,axiom,
    ! [X: rat] :
      aa_rat_rat(abs_abs_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X),X) ).

% abs_if_raw
tff(fact_6864_abs__if__raw,axiom,
    ! [X: code_integer] :
      abs_abs_Code_integer(X) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),X) ).

% abs_if_raw
tff(fact_6865_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_6866_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_6867_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_6868_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_6869_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_6870_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_6871_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_6872_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_6873_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_6874_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6875_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6876_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6877_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_6878_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_6879_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_6880_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_6881_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6882_abs__diff__le__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6883_abs__diff__le__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6884_abs__diff__le__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6885_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6886_abs__diff__less__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6887_abs__diff__less__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6888_abs__diff__less__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6889_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( finite_finite_int(A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: int] :
            ( member_int(X3,A2)
            & ( aa_int_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( member_int(Xa3,A2)
               => ( ( X3 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6890_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: complex] :
            ( member_complex(X3,A2)
            & ( aa_complex_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( member_complex(Xa3,A2)
               => ( ( X3 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6891_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & ( aa_nat_nat(F,X3) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A2)
               => ( ( X3 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_6892_sum__SucD,axiom,
    ! [F: fun_nat_nat,A2: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,Nb) )
     => ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X2)) ) ) ).

% sum_SucD
tff(fact_6893_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: fun_int_nat] :
      ( finite_finite_int(A2)
     => ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
      <=> ? [X3: int] :
            ( member_int(X3,A2)
            & ( aa_int_nat(F,X3) = one_one_nat )
            & ! [Xa3: int] :
                ( member_int(Xa3,A2)
               => ( ( X3 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6894_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(A2)
     => ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
      <=> ? [X3: complex] :
            ( member_complex(X3,A2)
            & ( aa_complex_nat(F,X3) = one_one_nat )
            & ! [Xa3: complex] :
                ( member_complex(Xa3,A2)
               => ( ( X3 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6895_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
      <=> ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A2)
            & ( aa_nat_nat(F,X3) = one_one_nat )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A2)
               => ( ( X3 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_6896_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_6897_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_6898_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_6899_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_6900_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_6901_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_6902_power__minus__Bit1,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6903_power__minus__Bit1,axiom,
    ! [Xb: int,K: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6904_power__minus__Bit1,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6905_power__minus__Bit1,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6906_power__minus__Bit1,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6907_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6908_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6909_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6910_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_6911_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_6912_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_6913_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_6914_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_6915_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_6916_lemma__interval__lt,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [Y2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y2))),D3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),B) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_6917_sum__power__add,axiom,
    ! [Xb: complex,Mb: nat,I6: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ls(Xb),Mb)),I6) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),I6)) ).

% sum_power_add
tff(fact_6918_sum__power__add,axiom,
    ! [Xb: rat,Mb: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_lt(Xb),Mb),I6) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),I6)) ).

% sum_power_add
tff(fact_6919_sum__power__add,axiom,
    ! [Xb: int,Mb: nat,I6: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_lu(Xb),Mb),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Mb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),I6)) ).

% sum_power_add
tff(fact_6920_sum__power__add,axiom,
    ! [Xb: real,Mb: nat,I6: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lv(Xb),Mb)),I6) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),I6)) ).

% sum_power_add
tff(fact_6921_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Nb,Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lw(G),Nb),Mb)),set_or1269000886237332187st_nat(Nb,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_6922_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Nb,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lx(G),Nb),Mb)),set_or1269000886237332187st_nat(Nb,Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_6923_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_6924_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_6925_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_6926_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_6927_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_6928_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6929_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6930_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6931_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6932_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_6933_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_af,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cf,Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_6934_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_af,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_de(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_6935_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xb))) ).

% abs_add_one_gt_zero
tff(fact_6936_abs__add__one__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6937_abs__add__one__gt__zero,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6938_abs__add__one__gt__zero,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6939_sum__diff__nat,axiom,
    ! [B3: set_complex,A2: set_complex,F: fun_complex_nat] :
      ( finite3207457112153483333omplex(B3)
     => ( ord_le211207098394363844omplex(B3,A2)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B3)) ) ) ) ).

% sum_diff_nat
tff(fact_6940_sum__diff__nat,axiom,
    ! [B3: set_int,A2: set_int,F: fun_int_nat] :
      ( finite_finite_int(B3)
     => ( ord_less_eq_set_int(B3,A2)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B3)) ) ) ) ).

% sum_diff_nat
tff(fact_6941_sum__diff__nat,axiom,
    ! [B3: set_nat,A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B3),A2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,B3)) = 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),B3)) ) ) ) ).

% sum_diff_nat
tff(fact_6942_sum__shift__lb__Suc0__0,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_6943_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_6944_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_6945_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_6946_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_6947_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6948_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6949_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6950_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_6951_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6952_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6953_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6954_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_6955_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6956_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6957_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6958_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_6959_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_6960_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_6961_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6962_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6963_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6964_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),A)),A) ).

% power3_eq_cube
tff(fact_6965_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),A) ).

% power3_eq_cube
tff(fact_6966_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),A)),A) ).

% power3_eq_cube
tff(fact_6967_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_6968_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_6969_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_6970_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_6971_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_6972_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_6973_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_6974_lemma__interval,axiom,
    ! [A: real,Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [Y2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y2))),D3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_6975_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_ly(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6976_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_lz(G),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6977_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lo(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6978_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(G)),set_or1269000886237332187st_nat(Mb,Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_6979_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_ma(F),set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb))),aa_nat_rat(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_6980_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_mb(F),set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_6981_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mc(F)),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_6982_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_6983_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),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_6984_abs__le__square__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6985_abs__le__square__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),aa_rat_rat(abs_abs_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),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_6986_abs__le__square__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6987_abs__square__eq__1,axiom,
    ! [Xb: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xb) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6988_abs__square__eq__1,axiom,
    ! [Xb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,Xb) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6989_abs__square__eq__1,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xb) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_6990_abs__square__eq__1,axiom,
    ! [Xb: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6991_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_6992_power__even__abs,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_even_abs
tff(fact_6993_power__even__abs,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_even_abs
tff(fact_6994_power__even__abs,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_even_abs
tff(fact_6995_power__even__abs,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) ) ) ).

% power_even_abs
tff(fact_6996_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(Mb,Nb))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_6997_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(Mb,Nb))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_6998_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_6999_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Mb,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7000_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,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,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7001_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7002_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Mb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_7003_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,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,Nb))),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,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7004_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7005_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,one),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Nb)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Nb),aa_num_Code_integer(numera6620942414471956472nteger,Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_7006_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_7007_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_7008_divmod__int__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_7009_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_md),Qr) ).

% Divides.adjust_div_def
tff(fact_7010_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xb: code_integer] :
      ( ! [X2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X2),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7011_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
         => aa_real_o(aa_real_fun_real_o(P,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7012_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xb: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X2)
         => aa_rat_o(aa_rat_fun_rat_o(P,X2),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,Xb)),aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7013_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: int] :
      ( ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X2)
         => aa_int_o(aa_int_fun_int_o(P,X2),aa_nat_int(aa_int_fun_nat_int(power_power_int,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7014_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7015_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7016_power2__le__iff__abs__le,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7017_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7018_abs__square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_7019_abs__square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_le_1
tff(fact_7020_abs__square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_7021_abs__square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_le_1
tff(fact_7022_abs__square__less__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_7023_abs__square__less__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_less_1
tff(fact_7024_abs__square__less__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_7025_abs__square__less__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_less_1
tff(fact_7026_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_7027_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_7028_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_def
tff(fact_7029_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_7030_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7031_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_7032_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_7033_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono_even
tff(fact_7034_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(Xb,I3)) )
     => ( ( groups6621422865394947399nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I3: complex] :
              ( member_complex(I3,I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_me(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7035_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xb,I3)) )
     => ( ( groups7713935264441627589nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I3: real] :
              ( member_real(I3,I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_mf(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7036_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xb,I3)) )
     => ( ( groups7501900531339628137nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_mg(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7037_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xb,I3)) )
     => ( ( groups7873554091576472773nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I3: int] :
              ( member_int(I3,I6)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I3),B))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_mh(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7038_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Xb,I3)) )
     => ( ( groups5808333547571424918x_real(Xb,I6) = one_one_real )
       => ( ! [I3: complex] :
              ( member_complex(I3,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(A,I3)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_mi(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7039_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xb,I3)) )
     => ( ( groups8097168146408367636l_real(Xb,I6) = one_one_real )
       => ( ! [I3: real] :
              ( member_real(I3,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A,I3)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_mj(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7040_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xb,I3)) )
     => ( ( groups8778361861064173332t_real(Xb,I6) = one_one_real )
       => ( ! [I3: int] :
              ( member_int(I3,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A,I3)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mk(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7041_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Xb,I3)) )
     => ( ( groups5058264527183730370ex_rat(Xb,I6) = one_one_rat )
       => ( ! [I3: complex] :
              ( member_complex(I3,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(A,I3)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ml(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7042_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xb,I3)) )
     => ( ( groups1300246762558778688al_rat(Xb,I6) = one_one_rat )
       => ( ! [I3: real] :
              ( member_real(I3,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I3)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_mm(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7043_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xb,I3)) )
     => ( ( groups2906978787729119204at_rat(Xb,I6) = one_one_rat )
       => ( ! [I3: nat] :
              ( aa_set_nat_o(member_nat(I3),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I3)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mn(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7044_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_mo(F)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(F,Mb)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_7045_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_mp(F),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_7046_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_mq(F),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_7047_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mr(F)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_7048_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_ms(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7049_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_mt(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7050_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mu(F)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7051_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_7052_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) = groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,Nb))) ).

% mask_eq_sum_exp
tff(fact_7053_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,Nb))) ).

% mask_eq_sum_exp
tff(fact_7054_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),set_or1269000886237332187st_nat(Mb,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7055_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7056_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),set_or1269000886237332187st_nat(Mb,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Mb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7057_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7058_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_mv(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7059_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_mw(G),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7060_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mx(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7061_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: 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))),Mb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_my(G)),set_or1269000886237332187st_nat(Mb,Nb)) ).

% sum.in_pairs
tff(fact_7062_eq__diff__eq_H,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z) ) ) ).

% eq_diff_eq'
tff(fact_7063_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_7064_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_7065_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_7066_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mz(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_7067_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_7068_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_7069_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_7070_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Mb)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_7071_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_7072_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,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_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_7073_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_7074_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_7075_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_7076_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_7077_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7078_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_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_ri631733984087533419it_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) ).

% signed_take_bit_numeral_bit1
tff(fact_7079_arctan__double,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),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,Xb)) = 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))),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_7080_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_7081_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_7082_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_7083_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_7084_sum__gp,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_complex,
        $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_7085_sum__gp,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_rat,
        $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ) ).

% sum_gp
tff(fact_7086_sum__gp,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_real,
        $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ).

% sum_gp
tff(fact_7087_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7088_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7089_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7090_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Mb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7091_split__part,axiom,
    ! [P: $o,Q: fun_int_fun_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_na((P)),Q)),X)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X) ) ) ).

% split_part
tff(fact_7092_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_7093_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_7094_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_7095_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_7096_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_7097_negative__zle,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zle
tff(fact_7098_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_7099_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_7100_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_7101_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_7102_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7103_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7104_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7105_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7106_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7107_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Mb) = zero_z3403309356797280102nteger )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7108_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7109_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7110_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7111_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7112_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7113_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7114_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_7115_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_7116_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_7117_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_7118_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_7119_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_7120_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_7121_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% of_nat_numeral
tff(fact_7122_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_7123_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% of_nat_numeral
tff(fact_7124_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_7125_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_Code_integer(numera6620942414471956472nteger,Nb) ).

% of_nat_numeral
tff(fact_7126_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7127_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7128_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7129_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7130_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7131_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7132_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7133_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7134_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7135_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7136_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_7137_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_7138_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_7139_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_7140_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_add
tff(fact_7141_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_7142_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_7143_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_7144_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_7145_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mult
tff(fact_7146_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7147_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7148_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7149_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7150_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Xb) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7151_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7152_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7153_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7154_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7155_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7156_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,Mb)),Nb) ).

% of_nat_power
tff(fact_7157_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb) ).

% of_nat_power
tff(fact_7158_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),Nb) ).

% of_nat_power
tff(fact_7159_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb) ).

% of_nat_power
tff(fact_7160_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),Nb) ).

% of_nat_power
tff(fact_7161_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_7162_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_7163_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_7164_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_7165_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_7166_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_7167_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7168_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7169_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7170_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7171_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7172_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7173_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7174_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7175_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7176_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7177_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7178_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = one_one_Code_integer )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7179_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_7180_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_7181_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_7182_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Nb = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_7183_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_arctan_iff
tff(fact_7184_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_7185_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_7186_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_7187_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_7188_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_7189_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n3304061248610475627l_real((P)) ).

% of_nat_of_bool
tff(fact_7190_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% of_nat_of_bool
tff(fact_7191_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_7192_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n356916108424825756nteger((P)) ).

% of_nat_of_bool
tff(fact_7193_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_7194_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_7195_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_7196_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_7197_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_7198_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7199_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_7200_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_7201_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_7202_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_nb(F),A2) ).

% of_nat_sum
tff(fact_7203_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_nc(F),A2) ).

% of_nat_sum
tff(fact_7204_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_nd(F),A2) ).

% of_nat_sum
tff(fact_7205_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups7501900531339628137nteger(aTP_Lamm_ne(F),A2) ).

% of_nat_sum
tff(fact_7206_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_nf(F)),A2) ).

% of_nat_sum
tff(fact_7207_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_ng(F)),A2) ).

% of_nat_sum
tff(fact_7208_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7209_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),zero_z3403309356797280102nteger)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7210_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7211_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7212_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7213_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).

% of_nat_Suc
tff(fact_7214_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_7215_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% of_nat_Suc
tff(fact_7216_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_7217_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).

% of_nat_Suc
tff(fact_7218_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Mb)) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)) ).

% of_nat_Suc
tff(fact_7219_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_7220_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_num_real(numeral_numeral_real,W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_7221_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_7222_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% less_numeral_Suc
tff(fact_7223_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_7224_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_7225_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% le_numeral_Suc
tff(fact_7226_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_7227_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_7228_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Nb) ).

% diff_numeral_Suc
tff(fact_7229_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_7230_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% max_numeral_Suc
tff(fact_7231_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_7232_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_7233_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_7234_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_7235_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_7236_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7237_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7238_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7239_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7240_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7241_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_7242_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_7243_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).

% dbl_dec_simps(1)
tff(fact_7244_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_7245_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_dec_simps(1)
tff(fact_7246_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_7247_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_7248_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).

% dbl_inc_simps(1)
tff(fact_7249_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_7250_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_inc_simps(1)
tff(fact_7251_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7252_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7253_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7254_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7255_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7256_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7257_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7258_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7259_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7260_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7261_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7262_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7263_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7264_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7265_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7266_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7267_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7268_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7269_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7270_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7271_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7272_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7273_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7274_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7275_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7276_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7277_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7278_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7279_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7280_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7281_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7282_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7283_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7284_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7285_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7286_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7287_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7288_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7289_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7290_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7291_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7292_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7293_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7294_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7295_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7296_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7297_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7298_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7299_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7300_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7301_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7302_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I2: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7303_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7304_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I2)),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7305_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7306_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7307_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7308_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7309_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7310_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7311_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_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))) ).

% signed_take_bit_numeral_bit0
tff(fact_7312_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_7313_Collect__case__prod__mono,axiom,
    ! [A2: fun_int_fun_int_o,B3: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A2,B3)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A2)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B3))) ) ).

% Collect_case_prod_mono
tff(fact_7314_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_nh),Prod) ).

% prod.disc_eq_case
tff(fact_7315_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_7316_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_7317_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_7318_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_7319_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)) ).

% mult_of_nat_commute
tff(fact_7320_arctan__monotone_H,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone'
tff(fact_7321_arctan__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% arctan_le_iff
tff(fact_7322_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_7323_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_7324_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_0_le_iff
tff(fact_7325_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_7326_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_7327_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_7328_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_7329_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_7330_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_7331_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_7332_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_7333_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_7334_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_7335_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_7336_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_7337_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_7338_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_7339_div__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_7340_div__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_7341_div__mult2__eq_H,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% div_mult2_eq'
tff(fact_7342_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7343_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7344_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7345_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7346_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7347_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7348_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7349_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7350_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7351_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7352_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I2)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_7353_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I2)),aa_nat_Code_integer(semiri4939895301339042750nteger,J)) ) ).

% of_nat_mono
tff(fact_7354_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I2)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_7355_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I2)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_7356_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_7357_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7358_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7359_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = divide6298287555418463151nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7360_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7361_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7362_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7363_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_7364_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7365_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_7366_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_7367_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_7368_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_7369_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_7370_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_7371_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_7372_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_7373_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_7374_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_7375_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_7376_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_7377_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mod
tff(fact_7378_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_7379_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_7380_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_7381_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_7382_int__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_nb(F),A2) ).

% int_sum
tff(fact_7383_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_nd(F),A2) ).

% int_sum
tff(fact_7384_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_7385_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,semiri4216267220026989637d_enat(Xb)),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_7386_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_7387_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_7388_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_7389_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ).

% of_nat_max
tff(fact_7390_nat__less__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_7391_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_7392_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7393_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_7394_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_7395_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7396_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Mb),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ) ).

% of_nat_diff
tff(fact_7397_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y2: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_7398_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7399_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_7400_dvd__imp__le__int,axiom,
    ! [I2: int,D: int] :
      ( ( I2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I2)) ) ) ).

% dvd_imp_le_int
tff(fact_7401_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_7402_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_7403_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_7404_int__zle__neg,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)))
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7405_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_7406_real__of__nat__div,axiom,
    ! [D: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_7407_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_7408_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_7409_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_7410_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_7411_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_7412_mod__mult2__eq_H,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb))) ).

% mod_mult2_eq'
tff(fact_7413_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_7414_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_7415_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_7416_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_7417_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_7418_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_7419_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_7420_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_7421_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_7422_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)),Mb)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7423_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_7424_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_7425_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7426_dbl__inc__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Xb)),one_one_complex) ).

% dbl_inc_def
tff(fact_7427_dbl__inc__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_inc_def
tff(fact_7428_dbl__inc__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xb) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_inc_def
tff(fact_7429_dbl__inc__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_inc_def
tff(fact_7430_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_7431_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_7432_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7433_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7434_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7435_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7436_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_7437_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_7438_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_abs_add_iff
tff(fact_7439_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_add_abs_iff
tff(fact_7440_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_7441_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_7442_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_7443_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_7444_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_7445_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_7446_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7447_dbl__dec__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Xb)),one_one_complex) ).

% dbl_dec_def
tff(fact_7448_dbl__dec__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_dec_def
tff(fact_7449_dbl__dec__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xb) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_dec_def
tff(fact_7450_dbl__dec__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_dec_def
tff(fact_7451_decr__lemma,axiom,
    ! [D: int,Xb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),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,Xb),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_7452_incr__lemma,axiom,
    ! [D: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),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,Xb),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_7453_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_7454_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_7455_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7456_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7457_double__arith__series,axiom,
    ! [A: complex,D: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ni(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),D))) ).

% double_arith_series
tff(fact_7458_double__arith__series,axiom,
    ! [A: rat,D: rat,Nb: 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_nj(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),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,Nb)),D))) ).

% double_arith_series
tff(fact_7459_double__arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nk(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D))) ).

% double_arith_series
tff(fact_7460_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_nl(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),D))) ).

% double_arith_series
tff(fact_7461_double__arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nm(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D))) ).

% double_arith_series
tff(fact_7462_double__arith__series,axiom,
    ! [A: real,D: real,Nb: 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_nn(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),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,Nb)),D))) ).

% double_arith_series
tff(fact_7463_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum
tff(fact_7464_double__gauss__sum,axiom,
    ! [Nb: 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,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum
tff(fact_7465_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum
tff(fact_7466_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_7467_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum
tff(fact_7468_double__gauss__sum,axiom,
    ! [Nb: 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,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_7469_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xb)),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,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_7470_arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_nk(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7471_arith__series,axiom,
    ! [A: code_integer,D: code_integer,Nb: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_nl(A),D),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),D))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7472_arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nm(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7473_gauss__sum,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7474_gauss__sum,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7475_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7476_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7477_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: 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),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7478_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7479_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7480_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7481_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: 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),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7482_sum__gp__offset,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb))))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ).

% sum_gp_offset
tff(fact_7483_sum__gp__offset,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_offset
tff(fact_7484_sum__gp__offset,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_7485_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_7486_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7487_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7488_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7489_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_no,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7490_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_np,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7491_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,produc6840382203811409530at_int(aTP_Lamm_nq,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7492_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_nr),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7493_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_ns),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7494_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,produc1830744345554046123nteger(aTP_Lamm_nt,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7495_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_7496_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_7497_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_nu,Xb)) ) ).

% monoseq_arctan_series
tff(fact_7498_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(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,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7499_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Nb))),H)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(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,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7500_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7501_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7502_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_nv,Xb)) ) ) ) ).

% ln_series
tff(fact_7503_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_kp(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7504_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_kr(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7505_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_7506_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_7507_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% monoseq_realpow
tff(fact_7508_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_7509_real__arch__simple,axiom,
    ! [Xb: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_7510_reals__Archimedean2,axiom,
    ! [Xb: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_7511_reals__Archimedean2,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_7512_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P,X_13)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_7513_arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arctan,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_nw,Xb)) ) ) ).

% arctan_series
tff(fact_7514_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7515_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7516_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_7517_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_7518_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7519_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7520_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7521_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7522_norm__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_7523_norm__le__zero__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_7524_suminf__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7525_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => ( suminf_complex(aa_com8181001469867462503omplex(power_power_complex,C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_7526_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_nx) = zero_zero_complex ).

% suminf_zero
tff(fact_7527_suminf__zero,axiom,
    suminf_real(aTP_Lamm_fb) = zero_zero_real ).

% suminf_zero
tff(fact_7528_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_bl) = zero_zero_nat ).

% suminf_zero
tff(fact_7529_suminf__zero,axiom,
    suminf_int(aTP_Lamm_ny) = zero_zero_int ).

% suminf_zero
tff(fact_7530_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_7531_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_7532_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7533_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7534_norm__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).

% norm_minus_commute
tff(fact_7535_norm__minus__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),A)) ).

% norm_minus_commute
tff(fact_7536_norm__ge__zero,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% norm_ge_zero
tff(fact_7537_norm__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_7538_norm__mult,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).

% norm_mult
tff(fact_7539_norm__divide,axiom,
    ! [A: real,B: 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)) ).

% norm_divide
tff(fact_7540_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_7541_sum__norm__le,axiom,
    ! [S3: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X2))),aa_real_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S3))),groups8097168146408367636l_real(G,S3)) ) ).

% sum_norm_le
tff(fact_7542_sum__norm__le,axiom,
    ! [S3: set_set_nat,F: fun_set_nat_complex,G: fun_set_nat_real] :
      ( ! [X2: set_nat] :
          ( member_set_nat(X2,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X2))),aa_set_nat_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S3))),groups5107569545109728110t_real(G,S3)) ) ).

% sum_norm_le
tff(fact_7543_sum__norm__le,axiom,
    ! [S3: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X2: int] :
          ( member_int(X2,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X2))),aa_int_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S3))),groups8778361861064173332t_real(G,S3)) ) ).

% sum_norm_le
tff(fact_7544_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X2))),aa_nat_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),S3))),aa_set_nat_real(groups6591440286371151544t_real(G),S3)) ) ).

% sum_norm_le
tff(fact_7545_sum__norm__le,axiom,
    ! [S3: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X2: complex] :
          ( member_complex(X2,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X2))),aa_complex_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S3))),groups5808333547571424918x_real(G,S3)) ) ).

% sum_norm_le
tff(fact_7546_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X2))),aa_nat_real(G,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),S3))),aa_set_nat_real(groups6591440286371151544t_real(G),S3)) ) ).

% sum_norm_le
tff(fact_7547_norm__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb) ).

% norm_power
tff(fact_7548_norm__power,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb) ).

% norm_power
tff(fact_7549_norm__sum,axiom,
    ! [F: fun_nat_complex,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nz(F)),A2)) ).

% norm_sum
tff(fact_7550_norm__sum,axiom,
    ! [F: fun_complex_complex,A2: set_complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A2))),groups5808333547571424918x_real(aTP_Lamm_oa(F),A2)) ).

% norm_sum
tff(fact_7551_norm__sum,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ob(F)),A2)) ).

% norm_sum
tff(fact_7552_norm__uminus__minus,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).

% norm_uminus_minus
tff(fact_7553_norm__uminus__minus,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) ).

% norm_uminus_minus
tff(fact_7554_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_7555_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(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_7556_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7557_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Nb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7558_norm__mult__less,axiom,
    ! [Xb: real,R2: real,Y: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2)) ) ) ).

% norm_mult_less
tff(fact_7559_norm__mult__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S2)) ) ) ).

% norm_mult_less
tff(fact_7560_norm__mult__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_mult_ineq
tff(fact_7561_norm__mult__ineq,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_mult_ineq
tff(fact_7562_norm__triangle__lt,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).

% norm_triangle_lt
tff(fact_7563_norm__triangle__lt,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),E) ) ).

% norm_triangle_lt
tff(fact_7564_norm__add__less,axiom,
    ! [Xb: real,R2: real,Y: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).

% norm_add_less
tff(fact_7565_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).

% norm_add_less
tff(fact_7566_norm__power__ineq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_7567_norm__power__ineq,axiom,
    ! [Xb: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_7568_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).

% norm_triangle_mono
tff(fact_7569_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S2)) ) ) ).

% norm_triangle_mono
tff(fact_7570_norm__triangle__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_7571_norm__triangle__ineq,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_7572_norm__triangle__le,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).

% norm_triangle_le
tff(fact_7573_norm__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),E) ) ).

% norm_triangle_le
tff(fact_7574_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_7575_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_7576_norm__diff__triangle__less,axiom,
    ! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_7577_norm__diff__triangle__less,axiom,
    ! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_7578_norm__triangle__sub,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)))) ).

% norm_triangle_sub
tff(fact_7579_norm__triangle__sub,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)))) ).

% norm_triangle_sub
tff(fact_7580_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))) ).

% norm_triangle_ineq4
tff(fact_7581_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))) ).

% norm_triangle_ineq4
tff(fact_7582_norm__diff__triangle__le,axiom,
    ! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7583_norm__diff__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7584_norm__triangle__le__diff,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E) ) ).

% norm_triangle_le_diff
tff(fact_7585_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y))),E) ) ).

% norm_triangle_le_diff
tff(fact_7586_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ).

% norm_diff_ineq
tff(fact_7587_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B))) ).

% norm_diff_ineq
tff(fact_7588_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% norm_triangle_ineq2
tff(fact_7589_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B))) ).

% norm_triangle_ineq2
tff(fact_7590_suminf__finite,axiom,
    ! [N2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => ( suminf_complex(F) = aa_set_nat_complex(groups2073611262835488442omplex(F),N2) ) ) ) ).

% suminf_finite
tff(fact_7591_suminf__finite,axiom,
    ! [N2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N2) ) ) ) ).

% suminf_finite
tff(fact_7592_suminf__finite,axiom,
    ! [N2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N2) ) ) ) ).

% suminf_finite
tff(fact_7593_suminf__finite,axiom,
    ! [N2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N2) ) ) ) ).

% suminf_finite
tff(fact_7594_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7595_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7596_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7597_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7598_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ).

% norm_triangle_ineq3
tff(fact_7599_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B))) ).

% norm_triangle_ineq3
tff(fact_7600_square__norm__one,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_7601_square__norm__one,axiom,
    ! [Xb: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_7602_norm__power__diff,axiom,
    ! [Z: real,W: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_7603_norm__power__diff,axiom,
    ! [Z: complex,W: complex,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_7604_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_oc) ).

% pi_series
tff(fact_7605_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Nb))),H)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,H),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oe(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7606_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Nb: 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)),Nb)),aa_nat_rat(power_power_rat(Z),Nb))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_og(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7607_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_oi(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7608_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_nw,Xb)) ) ).

% summable_arctan_series
tff(fact_7609_pred__subset__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ord_le2909477399203599485un_o_o(aTP_Lamm_oj(R),aTP_Lamm_oj(S3))
    <=> ord_le8980329558974975238eger_o(R,S3) ) ).

% pred_subset_eq2
tff(fact_7610_pred__subset__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ord_le3306795289226699318_num_o(aTP_Lamm_ok(R),aTP_Lamm_ok(S3))
    <=> ord_le880128212290418581um_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_7611_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aTP_Lamm_ol(R),aTP_Lamm_ol(S3))
    <=> ord_le8085105155179020875at_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_7612_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_om(R),aTP_Lamm_om(S3))
    <=> ord_le3146513528884898305at_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_7613_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_on(R),aTP_Lamm_on(S3))
    <=> ord_le2843351958646193337nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_7614_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ finite_finite_int(S3)
    <=> ! [M5: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M5),aa_int_int(abs_abs_int,N4))
          & member_int(N4,S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_7615_accp__subset,axiom,
    ! [R1: fun_Pr4799464687642144302_num_o,R22: fun_Pr4799464687642144302_num_o] :
      ( ord_le6421466325430216142_num_o(R1,R22)
     => ord_le3107050023622517222_num_o(accp_P3113834385874906142um_num(R22),accp_P3113834385874906142um_num(R1)) ) ).

% accp_subset
tff(fact_7616_accp__subset,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o] :
      ( ord_le3650406611588909774_nat_o(R1,R22)
     => aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(accp_P4275260045618599050at_nat(R22)),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_7617_accp__subset,axiom,
    ! [R1: fun_Pr9018117161326028078_int_o,R22: fun_Pr9018117161326028078_int_o] :
      ( ord_le1416746762259324110_int_o(R1,R22)
     => ord_le14110778710936002_int_o(accp_P1096762738010456898nt_int(R22),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_7618_accp__subset,axiom,
    ! [R1: fun_li1025265955262268170_nat_o,R22: fun_li1025265955262268170_nat_o] :
      ( ord_le4470944457981725354_nat_o(R1,R22)
     => ord_le4972187363741583025_nat_o(accp_list_nat(R22),accp_list_nat(R1)) ) ).

% accp_subset
tff(fact_7619_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(R1,R22)
     => ord_le7240017405068453697_nat_o(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_7620_lessThan__iff,axiom,
    ! [I2: set_nat,K: set_nat] :
      ( member_set_nat(I2,set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(ord_less_set_nat(I2),K) ) ).

% lessThan_iff
tff(fact_7621_lessThan__iff,axiom,
    ! [I2: rat,K: rat] :
      ( member_rat(I2,set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),K) ) ).

% lessThan_iff
tff(fact_7622_lessThan__iff,axiom,
    ! [I2: num,K: num] :
      ( member_num(I2,set_ord_lessThan_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I2),K) ) ).

% lessThan_iff
tff(fact_7623_lessThan__iff,axiom,
    ! [I2: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K) ) ).

% lessThan_iff
tff(fact_7624_lessThan__iff,axiom,
    ! [I2: int,K: int] :
      ( member_int(I2,aa_int_set_int(set_ord_lessThan_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K) ) ).

% lessThan_iff
tff(fact_7625_lessThan__iff,axiom,
    ! [I2: real,K: real] :
      ( member_real(I2,set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I2),K) ) ).

% lessThan_iff
tff(fact_7626_lessThan__iff,axiom,
    ! [I2: $o,K: $o] :
      ( member_o((I2),aa_o_set_o(set_ord_lessThan_o,(K)))
    <=> aa_o_o(ord_less_o((I2)),(K)) ) ).

% lessThan_iff
tff(fact_7627_summable__single,axiom,
    ! [I2: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fn(I2),F)) ).

% summable_single
tff(fact_7628_summable__single,axiom,
    ! [I2: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_oo(I2),F)) ).

% summable_single
tff(fact_7629_summable__single,axiom,
    ! [I2: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_op(I2),F)) ).

% summable_single
tff(fact_7630_summable__single,axiom,
    ! [I2: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_oq(I2),F)) ).

% summable_single
tff(fact_7631_summable__zero,axiom,
    summable_complex(aTP_Lamm_nx) ).

% summable_zero
tff(fact_7632_summable__zero,axiom,
    summable_real(aTP_Lamm_fb) ).

% summable_zero
tff(fact_7633_summable__zero,axiom,
    summable_nat(aTP_Lamm_bl) ).

% summable_zero
tff(fact_7634_summable__zero,axiom,
    summable_int(aTP_Lamm_ny) ).

% summable_zero
tff(fact_7635_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_lr(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7636_summable__iff__shift,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),K))
    <=> summable_complex(F) ) ).

% summable_iff_shift
tff(fact_7637_lessThan__subset__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_lessThan_rat(Xb),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_7638_lessThan__subset__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(Xb),set_ord_lessThan_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_7639_lessThan__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_7640_lessThan__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ord_less_eq_set_int(aa_int_set_int(set_ord_lessThan_int,Xb),aa_int_set_int(set_ord_lessThan_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_7641_lessThan__subset__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ord_less_eq_set_real(set_or5984915006950818249n_real(Xb),set_or5984915006950818249n_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_7642_lessThan__subset__iff,axiom,
    ! [Xb: $o,Y: $o] :
      ( ord_less_eq_set_o(aa_o_set_o(set_ord_lessThan_o,(Xb)),aa_o_set_o(set_ord_lessThan_o,(Y)))
    <=> aa_o_o(ord_less_eq_o((Xb)),(Y)) ) ).

% lessThan_subset_iff
tff(fact_7643_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_os(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7644_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7645_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ot(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7646_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ht(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7647_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ou(P),F)) ) ).

% summable_If_finite
tff(fact_7648_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ov(P),F)) ) ).

% summable_If_finite
tff(fact_7649_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ow(P),F)) ) ).

% summable_If_finite
tff(fact_7650_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ox(P),F)) ) ).

% summable_If_finite
tff(fact_7651_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oy(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7652_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_oz(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7653_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pa(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7654_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_pb(A2),F)) ) ).

% summable_If_finite_set
tff(fact_7655_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7656_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7657_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7658_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7659_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7660_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(power_power_complex,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7661_summable__norm__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ob(F))
     => summable_real(F) ) ).

% summable_norm_cancel
tff(fact_7662_summable__norm__cancel,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_nz(F))
     => summable_complex(F) ) ).

% summable_norm_cancel
tff(fact_7663_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_7664_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_7665_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_7666_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_7667_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_pc,C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7668_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_pd,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7669_summable__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_pe(F),C)) ) ).

% summable_mult2
tff(fact_7670_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_hg(F),C)) ) ).

% summable_mult2
tff(fact_7671_summable__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_pf(F),C)) ) ).

% summable_mult
tff(fact_7672_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_pg(F),C)) ) ).

% summable_mult
tff(fact_7673_summable__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ph(F),G)) ) ) ).

% summable_add
tff(fact_7674_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ho(F),G)) ) ) ).

% summable_add
tff(fact_7675_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hn(F),G)) ) ) ).

% summable_add
tff(fact_7676_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_pi(F),G)) ) ) ).

% summable_add
tff(fact_7677_summable__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pj(F),G)) ) ) ).

% summable_diff
tff(fact_7678_summable__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hr(F),G)) ) ) ).

% summable_diff
tff(fact_7679_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ot(F),C)) ) ).

% summable_divide
tff(fact_7680_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ht(F),C)) ) ).

% summable_divide
tff(fact_7681_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_lp(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_7682_summable__Suc__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_pk(F))
    <=> summable_complex(F) ) ).

% summable_Suc_iff
tff(fact_7683_summable__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => summable_real(aTP_Lamm_fy(F)) ) ).

% summable_minus
tff(fact_7684_summable__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_complex(aTP_Lamm_pl(F)) ) ).

% summable_minus
tff(fact_7685_summable__minus__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fy(F))
    <=> summable_real(F) ) ).

% summable_minus_iff
tff(fact_7686_summable__minus__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_pl(F))
    <=> summable_complex(F) ) ).

% summable_minus_iff
tff(fact_7687_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_lr(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7688_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(F)
     => summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7689_summable__sum,axiom,
    ! [I6: set_complex,F: fun_co8378349014452122532t_real] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => summable_real(aa_com20334259897932645t_real(F,I3)) )
     => summable_real(aa_fun471735231769099483t_real(aTP_Lamm_pn(I6),F)) ) ).

% summable_sum
tff(fact_7690_summable__sum,axiom,
    ! [I6: set_real,F: fun_re4238438383274954274t_real] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => summable_real(aa_real_fun_nat_real(F,I3)) )
     => summable_real(aa_fun628238740774635481t_real(aTP_Lamm_po(I6),F)) ) ).

% summable_sum
tff(fact_7691_summable__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_real] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => summable_real(aa_int_fun_nat_real(F,I3)) )
     => summable_real(aa_fun2842512154077127769t_real(aTP_Lamm_pq(I6),F)) ) ).

% summable_sum
tff(fact_7692_summable__sum,axiom,
    ! [I6: set_real,F: fun_re2084383175013312036omplex] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => summable_complex(aa_rea1531280491672885477omplex(F,I3)) )
     => summable_complex(aa_fun7632768522510439005omplex(aTP_Lamm_ps(I6),F)) ) ).

% summable_sum
tff(fact_7693_summable__sum,axiom,
    ! [I6: set_nat,F: fun_na1142814060743658952omplex] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => summable_complex(aa_nat6454746962644518025omplex(F,I3)) )
     => summable_complex(aa_fun4886358607025424513omplex(aTP_Lamm_pu(I6),F)) ) ).

% summable_sum
tff(fact_7694_summable__sum,axiom,
    ! [I6: set_int,F: fun_in8056368165505483812omplex] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => summable_complex(aa_int4144929030551567077omplex(F,I3)) )
     => summable_complex(aa_fun3543561798013398749omplex(aTP_Lamm_pw(I6),F)) ) ).

% summable_sum
tff(fact_7695_summable__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_int] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => summable_int(aa_int_fun_nat_int(F,I3)) )
     => summable_int(aa_fun8710705519803144921at_int(aTP_Lamm_py(I6),F)) ) ).

% summable_sum
tff(fact_7696_summable__sum,axiom,
    ! [I6: set_complex,F: fun_co3258106519541737638omplex] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => summable_complex(aa_com8181001469867462503omplex(F,I3)) )
     => summable_complex(aa_fun5003373910569599327omplex(aTP_Lamm_qa(I6),F)) ) ).

% summable_sum
tff(fact_7697_summable__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => summable_nat(aa_nat_fun_nat_nat(F,I3)) )
     => summable_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_qb(I6),F)) ) ).

% summable_sum
tff(fact_7698_summable__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => summable_real(aa_nat_fun_nat_real(F,I3)) )
     => summable_real(aa_fun7039663905532832253t_real(aTP_Lamm_qc(I6),F)) ) ).

% summable_sum
tff(fact_7699_suminf__le__const,axiom,
    ! [F: fun_nat_int,Xb: int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_7700_suminf__le__const,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_7701_suminf__le__const,axiom,
    ! [F: fun_nat_real,Xb: real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_7702_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ko(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_7703_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_qd(U)) ).

% lessThan_def
tff(fact_7704_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_qe,U)) ).

% lessThan_def
tff(fact_7705_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_qf,U)) ).

% lessThan_def
tff(fact_7706_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,U)) ).

% lessThan_def
tff(fact_7707_lessThan__def,axiom,
    ! [U: int] : aa_int_set_int(set_ord_lessThan_int,U) = collect_int(aa_int_fun_int_o(aTP_Lamm_qg,U)) ).

% lessThan_def
tff(fact_7708_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_qh,U)) ).

% lessThan_def
tff(fact_7709_lessThan__def,axiom,
    ! [U: $o] : aa_o_set_o(set_ord_lessThan_o,(U)) = collect_o(aTP_Lamm_qi((U))) ).

% lessThan_def
tff(fact_7710_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,Xb: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7711_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,Xb: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7712_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,Xb: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),Xb)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_7713_finite__nat__iff__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_7714_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [K2: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K2)) ) ).

% finite_nat_bounded
tff(fact_7715_powser__insidea,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_qk(F),Z)) ) ) ).

% powser_insidea
tff(fact_7716_powser__insidea,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_qm(F),Z)) ) ) ).

% powser_insidea
tff(fact_7717_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( summable_real(F)
       => ( summable_real(G)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),suminf_real(G)) ) ) ) ).

% suminf_le
tff(fact_7718_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G)) ) ) ) ).

% suminf_le
tff(fact_7719_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( summable_int(F)
       => ( summable_int(G)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),suminf_int(G)) ) ) ) ).

% suminf_le
tff(fact_7720_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(F)
     => ( suminf_complex(F) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),K))),aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_7721_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_lr(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_7722_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_complex,K: nat] :
      ( summable_complex(F)
     => ( suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),K)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(F)),aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_minus_initial_segment
tff(fact_7723_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_lr(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_7724_lessThan__strict__subset__iff,axiom,
    ! [Mb: rat,Nb: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(Mb),set_ord_lessThan_rat(Nb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7725_lessThan__strict__subset__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_less_set_num(set_ord_lessThan_num(Mb),set_ord_lessThan_num(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7726_lessThan__strict__subset__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7727_lessThan__strict__subset__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ord_less_set_int(aa_int_set_int(set_ord_lessThan_int,Mb),aa_int_set_int(set_ord_lessThan_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7728_lessThan__strict__subset__iff,axiom,
    ! [Mb: real,Nb: real] :
      ( ord_less_set_real(set_or5984915006950818249n_real(Mb),set_or5984915006950818249n_real(Nb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7729_lessThan__strict__subset__iff,axiom,
    ! [Mb: $o,Nb: $o] :
      ( ord_less_set_o(aa_o_set_o(set_ord_lessThan_o,(Mb)),aa_o_set_o(set_ord_lessThan_o,(Nb)))
    <=> aa_o_o(ord_less_o((Mb)),(Nb)) ) ).

% lessThan_strict_subset_iff
tff(fact_7730_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_os(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7731_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7732_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7733_summable__zero__power,axiom,
    summable_int(aa_int_fun_nat_int(power_power_int,zero_zero_int)) ).

% summable_zero_power
tff(fact_7734_summable__zero__power,axiom,
    summable_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex)) ).

% summable_zero_power
tff(fact_7735_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_7736_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Nb: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_7737_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_7738_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Nb: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_7739_suminf__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(F)),C) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_pe(F),C)) ) ) ).

% suminf_mult2
tff(fact_7740_suminf__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_hg(F),C)) ) ) ).

% suminf_mult2
tff(fact_7741_suminf__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_pf(F),C)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),suminf_complex(F)) ) ) ).

% suminf_mult
tff(fact_7742_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_pg(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_7743_suminf__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,suminf_complex(F)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ph(F),G)) ) ) ) ).

% suminf_add
tff(fact_7744_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_ho(F),G)) ) ) ) ).

% suminf_add
tff(fact_7745_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_hn(F),G)) ) ) ) ).

% suminf_add
tff(fact_7746_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_pi(F),G)) ) ) ) ).

% suminf_add
tff(fact_7747_suminf__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(F)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pj(F),G)) ) ) ) ).

% suminf_diff
tff(fact_7748_suminf__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_hr(F),G)) ) ) ) ).

% suminf_diff
tff(fact_7749_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ot(F),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_7750_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ht(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_7751_suminf__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_fy(F)) = aa_real_real(uminus_uminus_real,suminf_real(F)) ) ) ).

% suminf_minus
tff(fact_7752_suminf__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_pl(F)) = aa_complex_complex(uminus1482373934393186551omplex,suminf_complex(F)) ) ) ).

% suminf_minus
tff(fact_7753_suminf__sum,axiom,
    ! [I6: set_complex,F: fun_co8378349014452122532t_real] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => summable_real(aa_com20334259897932645t_real(F,I3)) )
     => ( suminf_real(aa_fun471735231769099483t_real(aTP_Lamm_pn(I6),F)) = groups5808333547571424918x_real(aTP_Lamm_qn(F),I6) ) ) ).

% suminf_sum
tff(fact_7754_suminf__sum,axiom,
    ! [I6: set_real,F: fun_re4238438383274954274t_real] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => summable_real(aa_real_fun_nat_real(F,I3)) )
     => ( suminf_real(aa_fun628238740774635481t_real(aTP_Lamm_po(I6),F)) = groups8097168146408367636l_real(aTP_Lamm_qo(F),I6) ) ) ).

% suminf_sum
tff(fact_7755_suminf__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_real] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => summable_real(aa_int_fun_nat_real(F,I3)) )
     => ( suminf_real(aa_fun2842512154077127769t_real(aTP_Lamm_pq(I6),F)) = groups8778361861064173332t_real(aTP_Lamm_qp(F),I6) ) ) ).

% suminf_sum
tff(fact_7756_suminf__sum,axiom,
    ! [I6: set_real,F: fun_re2084383175013312036omplex] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => summable_complex(aa_rea1531280491672885477omplex(F,I3)) )
     => ( suminf_complex(aa_fun7632768522510439005omplex(aTP_Lamm_ps(I6),F)) = groups5754745047067104278omplex(aTP_Lamm_qq(F),I6) ) ) ).

% suminf_sum
tff(fact_7757_suminf__sum,axiom,
    ! [I6: set_nat,F: fun_na1142814060743658952omplex] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => summable_complex(aa_nat6454746962644518025omplex(F,I3)) )
     => ( suminf_complex(aa_fun4886358607025424513omplex(aTP_Lamm_pu(I6),F)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_qr(F)),I6) ) ) ).

% suminf_sum
tff(fact_7758_suminf__sum,axiom,
    ! [I6: set_int,F: fun_in8056368165505483812omplex] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => summable_complex(aa_int4144929030551567077omplex(F,I3)) )
     => ( suminf_complex(aa_fun3543561798013398749omplex(aTP_Lamm_pw(I6),F)) = groups3049146728041665814omplex(aTP_Lamm_qs(F),I6) ) ) ).

% suminf_sum
tff(fact_7759_suminf__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_int] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => summable_int(aa_int_fun_nat_int(F,I3)) )
     => ( suminf_int(aa_fun8710705519803144921at_int(aTP_Lamm_py(I6),F)) = groups4538972089207619220nt_int(aTP_Lamm_qt(F),I6) ) ) ).

% suminf_sum
tff(fact_7760_suminf__sum,axiom,
    ! [I6: set_complex,F: fun_co3258106519541737638omplex] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => summable_complex(aa_com8181001469867462503omplex(F,I3)) )
     => ( suminf_complex(aa_fun5003373910569599327omplex(aTP_Lamm_qa(I6),F)) = groups7754918857620584856omplex(aTP_Lamm_qu(F),I6) ) ) ).

% suminf_sum
tff(fact_7761_suminf__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => summable_nat(aa_nat_fun_nat_nat(F,I3)) )
     => ( suminf_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_qb(I6),F)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_qv(F)),I6) ) ) ).

% suminf_sum
tff(fact_7762_suminf__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => summable_real(aa_nat_fun_nat_real(F,I3)) )
     => ( suminf_real(aa_fun7039663905532832253t_real(aTP_Lamm_qc(I6),F)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qw(F)),I6) ) ) ).

% suminf_sum
tff(fact_7763_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Nb: nat,I2: nat] :
      ( summable_int(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I2))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7764_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Nb: nat,I2: nat] :
      ( summable_nat(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7765_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Nb: nat,I2: nat] :
      ( summable_real(F)
     => ( ! [M: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I2))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_7766_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N4: nat] : aa_nat_real(F,N4) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7767_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N4: nat] : aa_nat_nat(F,N4) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7768_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N4: nat] : aa_nat_int(F,N4) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7769_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_nonneg
tff(fact_7770_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_nonneg
tff(fact_7771_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_nonneg
tff(fact_7772_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_pos
tff(fact_7773_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_pos
tff(fact_7774_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_pos
tff(fact_7775_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_kp(F)) ).

% summable_0_powser
tff(fact_7776_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_kr(F)) ).

% summable_0_powser
tff(fact_7777_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_kp(F)) ).

% summable_zero_power'
tff(fact_7778_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_kr(F)) ).

% summable_zero_power'
tff(fact_7779_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_qx(F)) ).

% summable_zero_power'
tff(fact_7780_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qy(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7781_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_qz(F),Z)) ) ).

% powser_split_head(3)
tff(fact_7782_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_qy(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7783_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qz(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z)) ) ).

% summable_powser_split_head
tff(fact_7784_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ra(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7785_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,Mb: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rb(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7786_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_nz(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7787_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_ko(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7788_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_7789_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_rd(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_7790_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ko(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_ko(F))) ) ).

% summable_rabs
tff(fact_7791_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: fun_int_nat,Nb: int] :
      ( ! [X2: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Q,X2)),aa_int_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,aa_int_set_int(set_ord_lessThan_int,Nb))),groups4541462559716669496nt_nat(Q,aa_int_set_int(set_ord_lessThan_int,Nb))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lm(Q),P),aa_int_set_int(set_ord_lessThan_int,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_7792_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Nb: real] :
      ( ! [X2: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X2)),aa_real_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Nb))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Nb))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_lk(Q),P),set_or5984915006950818249n_real(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_7793_sum__diff__distrib,axiom,
    ! [Q: fun_o_nat,P: fun_o_nat,Nb: $o] :
      ( ! [X2: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Q,(X2))),aa_o_nat(P,(X2)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8507830703676809646_o_nat(P,aa_o_set_o(set_ord_lessThan_o,(Nb)))),groups8507830703676809646_o_nat(Q,aa_o_set_o(set_ord_lessThan_o,(Nb)))) = groups8507830703676809646_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_re(Q),P),aa_o_set_o(set_ord_lessThan_o,(Nb))) ) ) ).

% sum_diff_distrib
tff(fact_7794_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Nb: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X2)),aa_nat_nat(P,X2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ln(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_7795_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_7796_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_7797_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_7798_suminf__pos2,axiom,
    ! [F: fun_nat_real,I2: nat] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_7799_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I2: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_7800_suminf__pos2,axiom,
    ! [F: fun_nat_int,I2: nat] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I2))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_7801_powser__inside,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z)) ) ) ).

% powser_inside
tff(fact_7802_powser__inside,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z)) ) ) ).

% powser_inside
tff(fact_7803_summable__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_7804_summable__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,C)) ) ).

% summable_geometric
tff(fact_7805_complete__algebra__summable__geometric,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_7806_complete__algebra__summable__geometric,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_7807_suminf__split__head,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_pk(F)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(F)),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7808_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_lp(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_7809_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_7810_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_7811_pi__half__neq__two,axiom,
    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_neq_two
tff(fact_7812_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D3)))),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))),D3)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_7813_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ly(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7814_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_lz(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7815_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lo(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7816_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7817_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_rf(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7818_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_rg(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7819_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rh(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7820_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ma(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7821_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_mb(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7822_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mc(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7823_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,Nb: nat,R2: rat] : aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),R2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_ri(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_7824_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,Nb: nat,R2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rj(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_7825_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_Code_integer,Nb: nat,R2: code_integer] : minus_8373710615458151222nteger(groups7501900531339628137nteger(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),R2)) = groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_rk(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_7826_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,Nb: nat,R2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),R2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rl(F),R2)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_7827_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_ob(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_ob(F))) ) ).

% summable_norm
tff(fact_7828_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_nz(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_nz(F))) ) ).

% summable_norm
tff(fact_7829_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lo(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_7830_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_7831_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I6: set_nat] :
      ( summable_int(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,I6)),suminf_int(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7832_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I6: set_nat] :
      ( summable_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I6)),suminf_nat(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7833_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I6: set_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(finite_finite_nat,I6)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I6))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),I6)),suminf_real(F)) ) ) ) ).

% sum_le_suminf
tff(fact_7834_pred__equals__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ! [X3: code_integer,Xa3: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Xa3)),R)
        <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Xa3)),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7835_pred__equals__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ! [X3: num,Xa3: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X3,Xa3),R)
        <=> member7279096912039735102um_num(product_Pair_num_num(X3,Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7836_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X3: nat,Xa3: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X3,Xa3),R)
        <=> member9148766508732265716at_num(product_Pair_nat_num(X3,Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7837_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Xa3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3),R)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7838_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa3),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_7839_bot__empty__eq2,axiom,
    ! [X: code_integer,Xa: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(bot_bo897209107778963633un_o_o,X),(Xa))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Xa)),bot_bo5379713665208646970eger_o) ) ).

% bot_empty_eq2
tff(fact_7840_bot__empty__eq2,axiom,
    ! [X: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(bot_bo3657453288633029378_num_o,X),Xa)
    <=> member7279096912039735102um_num(product_Pair_num_num(X,Xa),bot_bo9056780473022590049um_num) ) ).

% bot_empty_eq2
tff(fact_7841_bot__empty__eq2,axiom,
    ! [X: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X),Xa)
    <=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_7842_bot__empty__eq2,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X),Xa)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_7843_bot__empty__eq2,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_7844_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_7845_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_7846_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_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_le_two
tff(fact_7847_one__diff__power__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7848_one__diff__power__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7849_one__diff__power__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7850_one__diff__power__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7851_power__diff__1__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),one_one_complex) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),one_one_complex)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7852_power__diff__1__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7853_power__diff__1__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7854_power__diff__1__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7855_geometric__sum,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( Xb != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),one_one_complex)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_7856_geometric__sum,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_7857_geometric__sum,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ) ).

% geometric_sum
tff(fact_7858_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_qy(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7859_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_qj(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_qz(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7860_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_qy(F),Z))),Z) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7861_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_qz(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7862_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_real(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_lr(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_7863_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( summable_complex(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_7864_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_complex,E: real] :
      ( summable_complex(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(M2,N7)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7865_summable__partial__sum__bound,axiom,
    ! [F: fun_nat_real,E: real] :
      ( summable_real(F)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ~ ! [N6: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M2)
                 => ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(M2,N7)))),E) ) ) ) ).

% summable_partial_sum_bound
tff(fact_7866_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7867_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R2),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M6)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_rm(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_7868_sum__gp__strict,axiom,
    ! [Xb: complex,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ).

% sum_gp_strict
tff(fact_7869_sum__gp__strict,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_strict
tff(fact_7870_sum__gp__strict,axiom,
    ! [Xb: real,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_strict
tff(fact_7871_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rn(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ro(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7872_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rp(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rq(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7873_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rr(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rs(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7874_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rt(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ru(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7875_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_7876_diff__power__eq__sum,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),aa_nat_nat(suc,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_rv(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7877_diff__power__eq__sum,axiom,
    ! [Xb: rat,Nb: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_rw(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7878_diff__power__eq__sum,axiom,
    ! [Xb: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_nat(suc,Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_rx(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7879_diff__power__eq__sum,axiom,
    ! [Xb: real,Nb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ry(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7880_power__diff__sumr2,axiom,
    ! [Xb: complex,Nb: nat,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_rz(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7881_power__diff__sumr2,axiom,
    ! [Xb: rat,Nb: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_sa(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7882_power__diff__sumr2,axiom,
    ! [Xb: int,Nb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_sb(Xb),Nb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7883_power__diff__sumr2,axiom,
    ! [Xb: real,Nb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sc(Xb),Nb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7884_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_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_ge_zero
tff(fact_7885_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_7886_summable__ratio__test,axiom,
    ! [C: real,N2: nat,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_7887_summable__ratio__test,axiom,
    ! [C: real,N2: nat,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_7888_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_7889_arctan__one,axiom,
    aa_real_real(arctan,one_one_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)))) ).

% arctan_one
tff(fact_7890_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_Code_integer,K5: code_integer,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,P4)),K5) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),K5)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7891_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_rat,K5: rat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),K5) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7892_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),K5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7893_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),K5) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7894_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),K5) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K5)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7895_one__diff__power__eq_H,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_sd(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7896_one__diff__power__eq_H,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_se(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7897_one__diff__power__eq_H,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_sf(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7898_one__diff__power__eq_H,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sg(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7899_subrelI,axiom,
    ! [R2: set_Pr448751882837621926eger_o,S2: set_Pr448751882837621926eger_o] :
      ( ! [X2: code_integer,Y4: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X2,(Y4)),R2)
         => member1379723562493234055eger_o(produc6677183202524767010eger_o(X2,(Y4)),S2) )
     => ord_le8980329558974975238eger_o(R2,S2) ) ).

% subrelI
tff(fact_7900_subrelI,axiom,
    ! [R2: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num] :
      ( ! [X2: num,Y4: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X2,Y4),R2)
         => member7279096912039735102um_num(product_Pair_num_num(X2,Y4),S2) )
     => ord_le880128212290418581um_num(R2,S2) ) ).

% subrelI
tff(fact_7901_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ! [X2: nat,Y4: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X2,Y4),R2)
         => member9148766508732265716at_num(product_Pair_nat_num(X2,Y4),S2) )
     => ord_le8085105155179020875at_num(R2,S2) ) ).

% subrelI
tff(fact_7902_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y4: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4),S2) )
     => ord_le3146513528884898305at_nat(R2,S2) ) ).

% subrelI
tff(fact_7903_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y4: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y4),S2) )
     => ord_le2843351958646193337nt_int(R2,S2) ) ).

% subrelI
tff(fact_7904_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_7905_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_7906_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_7907_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_7908_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N7)
              & aa_set_nat_o(member_nat(N7),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_7909_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_7910_accp__subset__induct,axiom,
    ! [D2: fun_Pr6331313853612757382_num_o,R: fun_Pr4799464687642144302_num_o,Xb: product_prod_num_num,P: fun_Pr6331313853612757382_num_o] :
      ( ord_le3107050023622517222_num_o(D2,accp_P3113834385874906142um_num(R))
     => ( ! [X2: product_prod_num_num,Z3: product_prod_num_num] :
            ( aa_Pro7685007914727650815_num_o(D2,X2)
           => ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R,Z3),X2)
             => aa_Pro7685007914727650815_num_o(D2,Z3) ) )
       => ( aa_Pro7685007914727650815_num_o(D2,Xb)
         => ( ! [X2: product_prod_num_num] :
                ( aa_Pro7685007914727650815_num_o(D2,X2)
               => ( ! [Z4: product_prod_num_num] :
                      ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R,Z4),X2)
                     => aa_Pro7685007914727650815_num_o(P,Z4) )
                 => aa_Pro7685007914727650815_num_o(P,X2) ) )
           => aa_Pro7685007914727650815_num_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7911_accp__subset__induct,axiom,
    ! [D2: fun_Pr4796943543331071514_nat_o,R: fun_Pr2028404973800837934_nat_o,Xb: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(D2),accp_P4275260045618599050at_nat(R))
     => ( ! [X2: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D2,X2)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z3),X2)
             => aa_Pro6150637604445964947_nat_o(D2,Z3) ) )
       => ( aa_Pro6150637604445964947_nat_o(D2,Xb)
         => ( ! [X2: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D2,X2)
               => ( ! [Z4: product_prod_nat_nat] :
                      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z4),X2)
                     => aa_Pro6150637604445964947_nat_o(P,Z4) )
                 => aa_Pro6150637604445964947_nat_o(P,X2) ) )
           => aa_Pro6150637604445964947_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7912_accp__subset__induct,axiom,
    ! [D2: fun_Pr3238374608701176162_int_o,R: fun_Pr9018117161326028078_int_o,Xb: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( ord_le14110778710936002_int_o(D2,accp_P1096762738010456898nt_int(R))
     => ( ! [X2: product_prod_int_int,Z3: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D2,X2)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z3),X2)
             => aa_Pro4592068669816069595_int_o(D2,Z3) ) )
       => ( aa_Pro4592068669816069595_int_o(D2,Xb)
         => ( ! [X2: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D2,X2)
               => ( ! [Z4: product_prod_int_int] :
                      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z4),X2)
                     => aa_Pro4592068669816069595_int_o(P,Z4) )
                 => aa_Pro4592068669816069595_int_o(P,X2) ) )
           => aa_Pro4592068669816069595_int_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7913_accp__subset__induct,axiom,
    ! [D2: fun_list_nat_o,R: fun_li1025265955262268170_nat_o,Xb: list_nat,P: fun_list_nat_o] :
      ( ord_le4972187363741583025_nat_o(D2,accp_list_nat(R))
     => ( ! [X2: list_nat,Z3: list_nat] :
            ( aa_list_nat_o(D2,X2)
           => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(R,Z3),X2)
             => aa_list_nat_o(D2,Z3) ) )
       => ( aa_list_nat_o(D2,Xb)
         => ( ! [X2: list_nat] :
                ( aa_list_nat_o(D2,X2)
               => ( ! [Z4: list_nat] :
                      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(R,Z4),X2)
                     => aa_list_nat_o(P,Z4) )
                 => aa_list_nat_o(P,X2) ) )
           => aa_list_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7914_accp__subset__induct,axiom,
    ! [D2: fun_nat_o,R: fun_nat_fun_nat_o,Xb: nat,P: fun_nat_o] :
      ( ord_le7240017405068453697_nat_o(D2,accp_nat(R))
     => ( ! [X2: nat,Z3: nat] :
            ( aa_nat_o(D2,X2)
           => ( aa_nat_o(aa_nat_fun_nat_o(R,Z3),X2)
             => aa_nat_o(D2,Z3) ) )
       => ( aa_nat_o(D2,Xb)
         => ( ! [X2: nat] :
                ( aa_nat_o(D2,X2)
               => ( ! [Z4: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(R,Z4),X2)
                     => aa_nat_o(P,Z4) )
                 => aa_nat_o(P,X2) ) )
           => aa_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_7915_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sh(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_si(F)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sj(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_7916_pred__subset__eq,axiom,
    ! [R: set_complex,S3: set_complex] :
      ( ord_le7644322562001619171plex_o(aTP_Lamm_ab(R),aTP_Lamm_ab(S3))
    <=> ord_le211207098394363844omplex(R,S3) ) ).

% pred_subset_eq
tff(fact_7917_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_a(R),aTP_Lamm_a(S3))
    <=> ord_less_eq_set_real(R,S3) ) ).

% pred_subset_eq
tff(fact_7918_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S3))
    <=> ord_le6893508408891458716et_nat(R,S3) ) ).

% pred_subset_eq
tff(fact_7919_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S3))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R),S3) ) ).

% pred_subset_eq
tff(fact_7920_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ae(R),aTP_Lamm_ae(S3))
    <=> ord_less_eq_set_int(R,S3) ) ).

% pred_subset_eq
tff(fact_7921_machin__Euler,axiom,
    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(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_7922_machin,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)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_7923_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = $ite(
                      (A3),
                      some_nat(zero_zero_nat),
                      $ite((B2),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_7924_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xb)
       => ( ! [A3: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A3),(B2)) )
             => ( ( Y = $ite(
                      (B2),
                      some_nat(one_one_nat),
                      $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A3),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_7925_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lo(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_7926_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_7927_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_num_num,A1: nat,A22: nat,A32: num,P: fun_fu8148986511470497476_num_o] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(A0,produc1195630363706982562at_num(A1,product_Pair_nat_num(A22,A32))))
     => ( ! [F2: fun_nat_fun_num_num,A3: nat,B2: nat,Acc: num] :
            ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F2,produc1195630363706982562at_num(A3,product_Pair_nat_num(B2,Acc))))
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A3)
               => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),one_one_nat)),B2),aa_num_num(aa_nat_fun_num_num(F2,A3),Acc)) )
             => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,F2),A3),B2),Acc) ) )
       => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7928_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_nat_nat,A1: nat,A22: nat,A32: nat,P: fun_fu2571895429041709858_nat_o] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(A0,produc487386426758144856at_nat(A1,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A22),A32))))
     => ( ! [F2: fun_nat_fun_nat_nat,A3: nat,B2: nat,Acc: nat] :
            ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Acc))))
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A3)
               => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),one_one_nat)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F2,A3),Acc)) )
             => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F2),A3),B2),Acc) ) )
       => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_7929_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
     => ( accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_P5556105721700978146at_nat )
           => ( ( Y = none_P5556105721700978146at_nat )
             => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Xba))) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(V2) )
               => ( ( Xba = none_P5556105721700978146at_nat )
                 => ( ( Y = none_P5556105721700978146at_nat )
                   => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))) ) ) )
           => ~ ! [A3: product_prod_nat_nat] :
                  ( ( Xaa = some_P7363390416028606310at_nat(A3) )
                 => ! [B2: product_prod_nat_nat] :
                      ( ( Xba = some_P7363390416028606310at_nat(B2) )
                     => ( ( Y = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A3),B2)) )
                       => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7930_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xb,Xaa,Xba) = Y )
     => ( accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(Xaa,Xba)))
       => ( ( ( Xaa = none_num )
           => ( ( Y = none_num )
             => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(none_num,Xba))) ) )
         => ( ! [V2: num] :
                ( ( Xaa = some_num(V2) )
               => ( ( Xba = none_num )
                 => ( ( Y = none_num )
                   => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(some_num(V2),none_num))) ) ) )
           => ~ ! [A3: num] :
                  ( ( Xaa = some_num(A3) )
                 => ! [B2: num] :
                      ( ( Xba = some_num(B2) )
                     => ( ( Y = some_num(aa_num_num(aa_num_fun_num_num(Xb,A3),B2)) )
                       => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(some_num(A3),some_num(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7931_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
     => ( accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_nat )
           => ( ( Y = none_nat )
             => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(none_nat,Xba))) ) )
         => ( ! [V2: nat] :
                ( ( Xaa = some_nat(V2) )
               => ( ( Xba = none_nat )
                 => ( ( Y = none_nat )
                   => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(some_nat(V2),none_nat))) ) ) )
           => ~ ! [A3: nat] :
                  ( ( Xaa = some_nat(A3) )
                 => ! [B2: nat] :
                      ( ( Xba = some_nat(B2) )
                     => ( ( Y = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A3),B2)) )
                       => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(some_nat(A3),some_nat(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_7932_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sk),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_7933_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xb: fun_nat_fun_num_num,Xaa: nat,Xba: nat,Xc: num,Y: num] :
      ( ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(Xb,Xaa),Xba),Xc) = Y )
     => ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(Xb,produc1195630363706982562at_num(Xaa,product_Pair_nat_num(Xba,Xc))))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_num_num(aa_nat_fun_num_num(Xb,Xaa),Xc))) )
           => ~ accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(Xb,produc1195630363706982562at_num(Xaa,product_Pair_nat_num(Xba,Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7934_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xba),Xc))))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_7935_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_num_num,A: nat,B: nat,Acc2: num] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F,produc1195630363706982562at_num(A,product_Pair_nat_num(B,Acc2))))
     => ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_num_num(aa_nat_fun_num_num(F,A),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7936_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B),Acc2))))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_7937_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_7938_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7939_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_7940_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_7941_sin__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(sin_real,Xb) ).

% sin_pi_minus
tff(fact_7942_cos__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi
tff(fact_7943_cos__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi2
tff(fact_7944_sin__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi
tff(fact_7945_sin__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi2
tff(fact_7946_cos__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_pi_minus
tff(fact_7947_cos__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_minus_pi
tff(fact_7948_sin__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_minus_pi
tff(fact_7949_sin__cos__squared__add3,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Xb))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Xb))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_7950_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7951_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_7952_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_7953_cos__pi__half,axiom,
    aa_real_real(cos_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 ).

% cos_pi_half
tff(fact_7954_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7955_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_7956_cos__two__pi,axiom,
    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))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7957_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_7958_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_7959_cos__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).

% cos_2pi_minus
tff(fact_7960_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_7961_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_7962_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7963_sin__cos__squared__add,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7964_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7965_sin__cos__squared__add2,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7966_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_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))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7967_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_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))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7968_sin__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_2pi_minus
tff(fact_7969_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7970_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7971_sin__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_7972_cos__one__sin__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) = one_one_complex )
     => ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_7973_cos__one__sin__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
     => ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_7974_polar__Ex,axiom,
    ! [Xb: real,Y: real] :
    ? [R3: real,A3: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(cos_real,A3)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(sin_real,A3)) ) ) ).

% polar_Ex
tff(fact_7975_sin__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_diff
tff(fact_7976_cos__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_7977_cos__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_7978_sin__double,axiom,
    ! [Xb: complex] : aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,Xb))),aa_complex_complex(cos_complex,Xb)) ).

% sin_double
tff(fact_7979_sin__double,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,Xb))),aa_real_real(cos_real,Xb)) ).

% sin_double
tff(fact_7980_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y4)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
      & ( aa_real_real(sin_real,Y4) = aa_real_real(sin_real,Xb) )
      & ( aa_real_real(cos_real,Y4) = aa_real_real(cos_real,Xb) ) ) ).

% sincos_principal_value
tff(fact_7981_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_7982_sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),one_one_real) ).

% sin_le_one
tff(fact_7983_cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),one_one_real) ).

% cos_le_one
tff(fact_7984_abs__sin__x__le__abs__x,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),aa_real_real(abs_abs_real,Xb)) ).

% abs_sin_x_le_abs_x
tff(fact_7985_sin__cos__le1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_7986_cos__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7987_cos__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7988_sin__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7989_sin__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7990_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(sin_real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_7991_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_7992_sin__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,Xb)) ).

% sin_ge_minus_one
tff(fact_7993_cos__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xb) = aa_real_real(cos_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_7994_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_7995_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_7996_cos__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,Xb)) ).

% cos_ge_minus_one
tff(fact_7997_abs__sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),one_one_real) ).

% abs_sin_le_one
tff(fact_7998_abs__cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb))),one_one_real) ).

% abs_cos_le_one
tff(fact_7999_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8000_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_8001_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8002_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_8003_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8004_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_8005_cos__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8006_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_8007_sin__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8008_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_8009_sin__times__sin,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8010_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_8011_cos__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_8012_cos__double,axiom,
    ! [Xb: 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))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_8013_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_8014_cos__double__sin,axiom,
    ! [W: 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))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_8015_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_8016_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8017_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8018_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8019_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),pi)
            & ( Xb = aa_real_real(cos_real,T4) )
            & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_8020_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_8021_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_8022_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_8023_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_8024_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_8025_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y2: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y2) = zero_zero_real ) )
         => ( Y2 = X2 ) ) ) ).

% cos_is_zero
tff(fact_8026_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_8027_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_8028_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8029_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),pi)
            & ( aa_real_real(cos_real,X2) = Y )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
                  & ( aa_real_real(cos_real,Y2) = Y ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_8030_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T4) )
              & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_8031_sincos__total__2pi__le,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T4) )
          & ( Y = aa_real_real(sin_real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_8032_sincos__total__2pi,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T4) )
               => ( Y != aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8033_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8034_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8035_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_8036_cos__times__cos,axiom,
    ! [W: complex,Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8037_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_8038_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_8039_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8040_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8041_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_8042_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_8043_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8044_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8045_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8046_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_8047_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B: nat] : aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(A,B)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_sl(F),A),B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_8048_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_sm(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8049_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_sn(F),A),B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8050_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_so(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8051_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_sp(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8052_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_8053_cos__double__cos,axiom,
    ! [W: 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))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_8054_cos__treble__cos,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit1,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,Xb))) ).

% cos_treble_cos
tff(fact_8055_cos__treble__cos,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,Xb))) ).

% cos_treble_cos
tff(fact_8056_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8057_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_8058_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8059_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8060_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2)
            & aa_real_o(aa_real_fun_real_o(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))))
            & ( aa_real_real(sin_real,X2) = Y )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),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,Y2) = Y ) )
               => ( Y2 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_8061_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_8062_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_8063_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ( ? [X3: nat] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X3: nat] : Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_8064_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8065_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8066_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_8067_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_8068_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_8069_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sq,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T4),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,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8070_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sq,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T4),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,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8071_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sq,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T4),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,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8072_tan__double,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8073_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_8074_in__measure,axiom,
    ! [Xb: num,Y: num,F: fun_num_nat] :
      ( member7279096912039735102um_num(product_Pair_num_num(Xb,Y),measure_num(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,Xb)),aa_num_nat(F,Y)) ) ).

% in_measure
tff(fact_8075_in__measure,axiom,
    ! [Xb: nat,Y: nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),measure_nat(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Xb)),aa_nat_nat(F,Y)) ) ).

% in_measure
tff(fact_8076_in__measure,axiom,
    ! [Xb: int,Y: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y),measure_int(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,Xb)),aa_int_nat(F,Y)) ) ).

% in_measure
tff(fact_8077_tan__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(tan_real,Xb) ).

% tan_periodic_pi
tff(fact_8078_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_8079_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_8080_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_8081_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_8082_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_8083_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_8084_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_8085_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_8086_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_8087_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_8088_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_8089_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_8090_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_8091_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_8092_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_8093_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_8094_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_8095_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% fact_Suc
tff(fact_8096_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_8097_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_8098_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_8099_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_n
tff(fact_8100_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_nat
tff(fact_8101_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8102_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8103_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8104_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8105_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8106_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_8107_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_8108_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_8109_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_8110_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_8111_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_gt_zero
tff(fact_8112_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_8113_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_8114_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_8115_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).

% fact_not_neg
tff(fact_8116_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_8117_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_8118_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_8119_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_8120_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_8121_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_8122_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_8123_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_8124_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_8125_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_8126_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_8127_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_8128_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).

% fact_dvd
tff(fact_8129_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_8130_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_8131_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8132_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_8133_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_8134_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8135_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8136_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8137_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8138_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8139_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8140_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_8141_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Nb),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_8142_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_8143_fact__le__power,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8144_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8145_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8146_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8147_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8148_tan__def,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(sin_complex,X)),aa_complex_complex(cos_complex,X)) ).

% tan_def
tff(fact_8149_tan__def,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,X)) ).

% tan_def
tff(fact_8150_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_8151_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_8152_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_8153_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_8154_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_8155_fact__numeral,axiom,
    ! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8156_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8157_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8158_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8159_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8160_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_8161_tan__45,axiom,
    aa_real_real(tan_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))))) = one_one_real ).

% tan_45
tff(fact_8162_fact__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8163_fact__code,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8164_fact__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8165_fact__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8166_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8167_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8168_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8169_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_Code_integer(semiri3624122377584611663nteger,Mb) = $ite(Mb = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8170_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8171_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8172_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8173_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8174_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8175_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8176_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8177_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_gt_zero
tff(fact_8178_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(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_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X2)) ) ) ).

% lemma_tan_total
tff(fact_8179_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(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 )
      & ! [Y2: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),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,Y2) = Y ) )
         => ( Y2 = X2 ) ) ) ).

% tan_total
tff(fact_8180_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_8181_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8182_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8183_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(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 ) ) ).

% lemma_tan_total1
tff(fact_8184_tan__minus__45,axiom,
    aa_real_real(tan_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)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_8185_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_8186_add__tan__eq,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_8187_add__tan__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_8188_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_8189_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(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 ) ) ) ).

% tan_total_pos
tff(fact_8190_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_8191_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8192_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_8193_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xb))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_8194_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_8195_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_8196_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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,Xb) = Y )
         => ( aa_real_real(arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_8197_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_sr(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_8198_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ss(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8199_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_rat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_st(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8200_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_su(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8201_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_int_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_sv(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8202_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_sw(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B7),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_8203_tan__add,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_8204_tan__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_8205_tan__diff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8206_tan__diff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_8207_lemma__tan__add1,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8208_lemma__tan__add1,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_8209_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),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,Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_8210_tan__half,axiom,
    ! [Xb: complex] : aa_complex_complex(tan_complex,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),one_one_complex)) ).

% tan_half
tff(fact_8211_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_8212_cos__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(cos_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X)),zero_zero_real) ).

% cos_coeff_def
tff(fact_8213_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sx,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T4),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,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8214_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sx,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T4),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,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8215_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sx,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T4),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,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8216_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] : aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sx,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,T4),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,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_8217_sin__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(sin_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_8218_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_8219_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_8220_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_8221_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_8222_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_8223_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_8224_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_8225_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_8226_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_8227_sin__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),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,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8228_cos__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),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,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8229_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(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,T4)),aa_real_real(sin_real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8230_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sy,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8231_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_sz,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_8232_real__sqrt__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,Xb) = aa_real_real(sqrt,Y) )
    <=> ( Xb = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_8233_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_8234_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_8235_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_8236_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% real_sqrt_le_iff
tff(fact_8237_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_8238_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_8239_exp__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% exp_le_cancel_iff
tff(fact_8240_sums__zero,axiom,
    aa_complex_o(sums_complex(aTP_Lamm_nx),zero_zero_complex) ).

% sums_zero
tff(fact_8241_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_fb),zero_zero_real) ).

% sums_zero
tff(fact_8242_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_bl),zero_zero_nat) ).

% sums_zero
tff(fact_8243_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_ny),zero_zero_int) ).

% sums_zero
tff(fact_8244_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_8245_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_8246_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_8247_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_8248_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_8249_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_8250_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_8251_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_8252_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_8253_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_8254_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_8255_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs2
tff(fact_8256_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_8257_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% one_le_exp_iff
tff(fact_8258_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_8259_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_kp(A)),Xb)
    <=> ( aa_nat_complex(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_8260_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_kr(A)),Xb)
    <=> ( aa_nat_real(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_8261_real__sqrt__abs,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs
tff(fact_8262_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_8263_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_8264_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_8265_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_8266_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_8267_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S2: real,Ta: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( aa_real_o(sums_real(F),S2)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S2),Ta) ) ) ) ).

% sums_le
tff(fact_8268_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S2: nat,Ta: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( aa_nat_o(sums_nat(F),S2)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S2),Ta) ) ) ) ).

% sums_le
tff(fact_8269_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S2: int,Ta: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( aa_int_o(sums_int(F),S2)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S2),Ta) ) ) ) ).

% sums_le
tff(fact_8270_norm__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Xb))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,Xb))) ).

% norm_exp
tff(fact_8271_norm__exp,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Xb))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,Xb))) ).

% norm_exp
tff(fact_8272_real__sqrt__minus,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,Xb)) ).

% real_sqrt_minus
tff(fact_8273_real__sqrt__power,axiom,
    ! [Xb: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),K) ).

% real_sqrt_power
tff(fact_8274_real__sqrt__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_8275_real__sqrt__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).

% real_sqrt_divide
tff(fact_8276_exp__times__arg__commute,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(exp_complex,A2)),A2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_complex_complex(exp_complex,A2)) ).

% exp_times_arg_commute
tff(fact_8277_exp__times__arg__commute,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(exp_real,A2)) ).

% exp_times_arg_commute
tff(fact_8278_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ).

% complex_diff
tff(fact_8279_sums__single,axiom,
    ! [I2: nat,F: fun_nat_complex] : aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fn(I2),F)),aa_nat_complex(F,I2)) ).

% sums_single
tff(fact_8280_sums__single,axiom,
    ! [I2: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oo(I2),F)),aa_nat_real(F,I2)) ).

% sums_single
tff(fact_8281_sums__single,axiom,
    ! [I2: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_op(I2),F)),aa_nat_nat(F,I2)) ).

% sums_single
tff(fact_8282_sums__single,axiom,
    ! [I2: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_oq(I2),F)),aa_nat_int(F,I2)) ).

% sums_single
tff(fact_8283_sums__mult,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_pf(F),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),A)) ) ).

% sums_mult
tff(fact_8284_sums__mult,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_pg(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ) ).

% sums_mult
tff(fact_8285_sums__mult2,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_pe(F),C)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),C)) ) ).

% sums_mult2
tff(fact_8286_sums__mult2,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_hg(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ).

% sums_mult2
tff(fact_8287_sums__add,axiom,
    ! [F: fun_nat_complex,A: complex,G: fun_nat_complex,B: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => ( aa_complex_o(sums_complex(G),B)
       => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ph(F),G)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)) ) ) ).

% sums_add
tff(fact_8288_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ho(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).

% sums_add
tff(fact_8289_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( aa_nat_o(sums_nat(F),A)
     => ( aa_nat_o(sums_nat(G),B)
       => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hn(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_8290_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( aa_int_o(sums_int(F),A)
     => ( aa_int_o(sums_int(G),B)
       => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_pi(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_8291_sums__diff,axiom,
    ! [F: fun_nat_complex,A: complex,G: fun_nat_complex,B: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => ( aa_complex_o(sums_complex(G),B)
       => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pj(F),G)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),B)) ) ) ).

% sums_diff
tff(fact_8292_sums__diff,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hr(F),G)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).

% sums_diff
tff(fact_8293_sums__divide,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ot(F),C)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),C)) ) ).

% sums_divide
tff(fact_8294_sums__divide,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ht(F),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ).

% sums_divide
tff(fact_8295_sums__minus,axiom,
    ! [F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aTP_Lamm_fy(F)),aa_real_real(uminus_uminus_real,A)) ) ).

% sums_minus
tff(fact_8296_sums__minus,axiom,
    ! [F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aTP_Lamm_pl(F)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ).

% sums_minus
tff(fact_8297_sums__sum,axiom,
    ! [I6: set_complex,F: fun_co8378349014452122532t_real,Xb: fun_complex_real] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => aa_real_o(sums_real(aa_com20334259897932645t_real(F,I3)),aa_complex_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun471735231769099483t_real(aTP_Lamm_pn(I6),F)),groups5808333547571424918x_real(Xb,I6)) ) ).

% sums_sum
tff(fact_8298_sums__sum,axiom,
    ! [I6: set_real,F: fun_re4238438383274954274t_real,Xb: fun_real_real] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => aa_real_o(sums_real(aa_real_fun_nat_real(F,I3)),aa_real_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun628238740774635481t_real(aTP_Lamm_po(I6),F)),groups8097168146408367636l_real(Xb,I6)) ) ).

% sums_sum
tff(fact_8299_sums__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_real,Xb: fun_int_real] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => aa_real_o(sums_real(aa_int_fun_nat_real(F,I3)),aa_int_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun2842512154077127769t_real(aTP_Lamm_pq(I6),F)),groups8778361861064173332t_real(Xb,I6)) ) ).

% sums_sum
tff(fact_8300_sums__sum,axiom,
    ! [I6: set_real,F: fun_re2084383175013312036omplex,Xb: fun_real_complex] :
      ( ! [I3: real] :
          ( member_real(I3,I6)
         => aa_complex_o(sums_complex(aa_rea1531280491672885477omplex(F,I3)),aa_real_complex(Xb,I3)) )
     => aa_complex_o(sums_complex(aa_fun7632768522510439005omplex(aTP_Lamm_ps(I6),F)),groups5754745047067104278omplex(Xb,I6)) ) ).

% sums_sum
tff(fact_8301_sums__sum,axiom,
    ! [I6: set_nat,F: fun_na1142814060743658952omplex,Xb: fun_nat_complex] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => aa_complex_o(sums_complex(aa_nat6454746962644518025omplex(F,I3)),aa_nat_complex(Xb,I3)) )
     => aa_complex_o(sums_complex(aa_fun4886358607025424513omplex(aTP_Lamm_pu(I6),F)),aa_set_nat_complex(groups2073611262835488442omplex(Xb),I6)) ) ).

% sums_sum
tff(fact_8302_sums__sum,axiom,
    ! [I6: set_int,F: fun_in8056368165505483812omplex,Xb: fun_int_complex] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => aa_complex_o(sums_complex(aa_int4144929030551567077omplex(F,I3)),aa_int_complex(Xb,I3)) )
     => aa_complex_o(sums_complex(aa_fun3543561798013398749omplex(aTP_Lamm_pw(I6),F)),groups3049146728041665814omplex(Xb,I6)) ) ).

% sums_sum
tff(fact_8303_sums__sum,axiom,
    ! [I6: set_int,F: fun_int_fun_nat_int,Xb: fun_int_int] :
      ( ! [I3: int] :
          ( member_int(I3,I6)
         => aa_int_o(sums_int(aa_int_fun_nat_int(F,I3)),aa_int_int(Xb,I3)) )
     => aa_int_o(sums_int(aa_fun8710705519803144921at_int(aTP_Lamm_py(I6),F)),groups4538972089207619220nt_int(Xb,I6)) ) ).

% sums_sum
tff(fact_8304_sums__sum,axiom,
    ! [I6: set_complex,F: fun_co3258106519541737638omplex,Xb: fun_complex_complex] :
      ( ! [I3: complex] :
          ( member_complex(I3,I6)
         => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(F,I3)),aa_complex_complex(Xb,I3)) )
     => aa_complex_o(sums_complex(aa_fun5003373910569599327omplex(aTP_Lamm_qa(I6),F)),groups7754918857620584856omplex(Xb,I6)) ) ).

% sums_sum
tff(fact_8305_sums__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_nat,Xb: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => aa_nat_o(sums_nat(aa_nat_fun_nat_nat(F,I3)),aa_nat_nat(Xb,I3)) )
     => aa_nat_o(sums_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_qb(I6),F)),aa_set_nat_nat(groups3542108847815614940at_nat(Xb),I6)) ) ).

% sums_sum
tff(fact_8306_sums__sum,axiom,
    ! [I6: set_nat,F: fun_nat_fun_nat_real,Xb: fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I6)
         => aa_real_o(sums_real(aa_nat_fun_nat_real(F,I3)),aa_nat_real(Xb,I3)) )
     => aa_real_o(sums_real(aa_fun7039663905532832253t_real(aTP_Lamm_qc(I6),F)),aa_set_nat_real(groups6591440286371151544t_real(Xb),I6)) ) ).

% sums_sum
tff(fact_8307_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_8308_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_8309_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
       => ( Xb = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_8310_exp__ge__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_ge_zero
tff(fact_8311_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_8312_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_8313_exp__add__commuting,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Xb) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_8314_exp__add__commuting,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_8315_mult__exp__exp,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) ).

% mult_exp_exp
tff(fact_8316_mult__exp__exp,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).

% mult_exp_exp
tff(fact_8317_exp__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_8318_exp__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_8319_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_os(C),F)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_8320_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_8321_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ta(C),F)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,D),C))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8322_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_tb(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8323_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_8324_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ).

% complex_add
tff(fact_8325_complex__norm,axiom,
    ! [Xb: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_8326_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,Xb)) = aa_real_real(sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_8327_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_8328_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_os(C),F)),A)
     => ( ( C != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),C)) ) ) ).

% sums_mult_D
tff(fact_8329_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_hk(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% sums_mult_D
tff(fact_8330_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S2: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_complex_o(sums_complex(aTP_Lamm_pk(F)),S2)
       => aa_complex_o(sums_complex(F),S2) ) ) ).

% sums_Suc_imp
tff(fact_8331_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_lp(F)),S2)
       => aa_real_o(sums_real(F),S2) ) ) ).

% sums_Suc_imp
tff(fact_8332_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ).

% exp_ge_add_one_self
tff(fact_8333_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_8334_sums__Suc,axiom,
    ! [F: fun_nat_complex,L: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_pk(F)),L)
     => aa_complex_o(sums_complex(F),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,L),aa_nat_complex(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8335_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_lp(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8336_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_lo(F)),L)
     => aa_nat_o(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8337_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_lz(F)),L)
     => aa_int_o(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8338_sums__Suc__iff,axiom,
    ! [F: fun_nat_complex,S2: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_pk(F)),S2)
    <=> aa_complex_o(sums_complex(F),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,S2),aa_nat_complex(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_8339_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( aa_real_o(sums_real(aTP_Lamm_lp(F)),S2)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_8340_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_complex,S2: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ( aa_nat_complex(F,I3) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tc(Nb),F)),S2)
      <=> aa_complex_o(sums_complex(F),S2) ) ) ).

% sums_zero_iff_shift
tff(fact_8341_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,S2: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ( aa_nat_real(F,I3) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_td(Nb),F)),S2)
      <=> aa_real_o(sums_real(F),S2) ) ) ).

% sums_zero_iff_shift
tff(fact_8342_exp__minus__inverse,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) = one_one_real ).

% exp_minus_inverse
tff(fact_8343_exp__minus__inverse,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb))) = one_one_complex ).

% exp_minus_inverse
tff(fact_8344_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: complex] : aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),Xb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_8345_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_8346_exp__of__nat2__mult,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_8347_exp__of__nat2__mult,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_8348_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_8349_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oy(A2),F)),aa_set_nat_complex(groups2073611262835488442omplex(F),A2)) ) ).

% sums_If_finite_set
tff(fact_8350_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_pb(A2),F)),groups3539618377306564664at_int(F,A2)) ) ).

% sums_If_finite_set
tff(fact_8351_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pa(A2),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) ) ).

% sums_If_finite_set
tff(fact_8352_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_oz(A2),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A2)) ) ).

% sums_If_finite_set
tff(fact_8353_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ou(P),F)),aa_set_nat_complex(groups2073611262835488442omplex(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8354_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ox(P),F)),groups3539618377306564664at_int(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8355_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ow(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8356_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ov(P),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_8357_sums__finite,axiom,
    ! [N2: set_nat,F: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_complex(F,N) = zero_zero_complex ) )
       => aa_complex_o(sums_complex(F),aa_set_nat_complex(groups2073611262835488442omplex(F),N2)) ) ) ).

% sums_finite
tff(fact_8358_sums__finite,axiom,
    ! [N2: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => aa_int_o(sums_int(F),groups3539618377306564664at_int(F,N2)) ) ) ).

% sums_finite
tff(fact_8359_sums__finite,axiom,
    ! [N2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => aa_nat_o(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N2)) ) ) ).

% sums_finite
tff(fact_8360_sums__finite,axiom,
    ! [N2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N2)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => aa_real_o(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N2)) ) ) ).

% sums_finite
tff(fact_8361_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(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),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))) ).

% complex_mult
tff(fact_8362_powser__sums__if,axiom,
    ! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_te(Mb),Z)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Mb)) ).

% powser_sums_if
tff(fact_8363_powser__sums__if,axiom,
    ! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_tf(Mb),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)) ).

% powser_sums_if
tff(fact_8364_powser__sums__if,axiom,
    ! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_tg(Mb),Z)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Mb)) ).

% powser_sums_if
tff(fact_8365_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_8366_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_kp(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8367_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_kr(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8368_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_8369_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real))
          & ( aa_real_real(exp_real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_8370_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_8371_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_8372_sums__iff__shift,axiom,
    ! [F: fun_nat_complex,Nb: nat,S2: complex] :
      ( aa_complex_o(sums_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),Nb)),S2)
    <=> aa_complex_o(sums_complex(F),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,S2),aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_iff_shift
tff(fact_8373_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Nb: nat,S2: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_lr(F),Nb)),S2)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_iff_shift
tff(fact_8374_sums__split__initial__segment,axiom,
    ! [F: fun_nat_complex,S2: complex,Nb: nat] :
      ( aa_complex_o(sums_complex(F),S2)
     => aa_complex_o(sums_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),Nb)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,S2),aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_split_initial_segment
tff(fact_8375_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S2: real,Nb: nat] :
      ( aa_real_o(sums_real(F),S2)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_lr(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_split_initial_segment
tff(fact_8376_sums__iff__shift_H,axiom,
    ! [F: fun_nat_complex,Nb: nat,S2: complex] :
      ( aa_complex_o(sums_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(F),Nb)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,S2),aa_set_nat_complex(groups2073611262835488442omplex(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
    <=> aa_complex_o(sums_complex(F),S2) ) ).

% sums_iff_shift'
tff(fact_8377_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Nb: nat,S2: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_lr(F),Nb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S2),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
    <=> aa_real_o(sums_real(F),S2) ) ).

% sums_iff_shift'
tff(fact_8378_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_complex,S3: complex,A2: set_nat,S4: complex,F: fun_nat_complex] :
      ( aa_complex_o(sums_complex(G),S3)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( S4 = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,S3),aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_th(G),F)),A2)) )
         => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aa_set5115289136882354743omplex(aTP_Lamm_ti(G),A2),F)),S4) ) ) ) ).

% sums_If_finite_set'
tff(fact_8379_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S3: real,A2: set_nat,S4: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S3)
     => ( aa_set_nat_o(finite_finite_nat,A2)
       => ( ( S4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_tj(G),F)),A2)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_tk(G),A2),F)),S4) ) ) ) ).

% sums_If_finite_set'
tff(fact_8380_Complex__sum_H,axiom,
    ! [F: fun_nat_real,S2: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_tl(F)),S2) = aa_real_complex(complex2(aa_set_nat_real(groups6591440286371151544t_real(F),S2)),zero_zero_real) ).

% Complex_sum'
tff(fact_8381_Complex__sum_H,axiom,
    ! [F: fun_complex_real,S2: set_complex] : groups7754918857620584856omplex(aTP_Lamm_tm(F),S2) = aa_real_complex(complex2(groups5808333547571424918x_real(F,S2)),zero_zero_real) ).

% Complex_sum'
tff(fact_8382_real__less__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_8383_real__le__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_8384_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_8385_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_8386_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_8387_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_8388_tanh__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% tanh_altdef
tff(fact_8389_tanh__altdef,axiom,
    ! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).

% tanh_altdef
tff(fact_8390_real__sqrt__unique,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_8391_real__le__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_8392_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_8393_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_8394_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xb )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_8395_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_8396_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_8397_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_8398_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_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_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_8399_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_8400_cos__45,axiom,
    aa_real_real(cos_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))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_8401_sin__45,axiom,
    aa_real_real(sin_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))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_8402_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_8403_exp__double,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_8404_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_8405_geometric__sums,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C))) ) ).

% geometric_sums
tff(fact_8406_geometric__sums,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(power_power_complex,C)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),C))) ) ).

% geometric_sums
tff(fact_8407_real__less__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_8408_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_tn),one_one_real) ).

% power_half_series
tff(fact_8409_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_8410_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_8411_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_8412_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_8413_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_8414_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_8415_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_8416_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_8417_exp__bound__half,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_8418_exp__bound__half,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_8419_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_to(G)),Xb) ) ).

% sums_if'
tff(fact_8420_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_tp(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sums_if
tff(fact_8421_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_8422_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_8423_exp__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_8424_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8425_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8426_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_8427_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_8428_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_8429_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8430_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_8431_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_8432_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_8433_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_8434_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_8435_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,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(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_8436_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_8437_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sy,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_8438_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_8439_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_8440_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_8441_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_8442_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_8443_cos__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_tq,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_paired
tff(fact_8444_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_tr,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8445_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ts,Z)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8446_diffs__equiv,axiom,
    ! [C: fun_nat_complex,Xb: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tt(C),Xb))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_tu(C),Xb)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_tt(C),Xb))) ) ).

% diffs_equiv
tff(fact_8447_diffs__equiv,axiom,
    ! [C: fun_nat_real,Xb: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_tv(C),Xb))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_tw(C),Xb)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_tv(C),Xb))) ) ).

% diffs_equiv
tff(fact_8448_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_8449_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_8450_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_8451_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_8452_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_8453_binomial__n__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Nb) = one_one_nat ).

% binomial_n_n
tff(fact_8454_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_8455_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_8456_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_8457_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_8458_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8459_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_8460_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_8461_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_8462_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_8463_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_8464_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_8465_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_8466_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8467_Suc__times__binomial,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ).

% Suc_times_binomial
tff(fact_8468_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_8469_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_8470_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_8471_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_8472_diffs__minus,axiom,
    ! [C: fun_nat_real,X: nat] : aa_nat_real(diffs_real(aTP_Lamm_fy(C)),X) = aa_real_real(uminus_uminus_real,aa_nat_real(diffs_real(C),X)) ).

% diffs_minus
tff(fact_8473_diffs__minus,axiom,
    ! [C: fun_nat_int,X: nat] : aa_nat_int(diffs_int(aTP_Lamm_tx(C)),X) = aa_int_int(uminus_uminus_int,aa_nat_int(diffs_int(C),X)) ).

% diffs_minus
tff(fact_8474_diffs__minus,axiom,
    ! [C: fun_nat_complex,X: nat] : aa_nat_complex(diffs_complex(aTP_Lamm_pl(C)),X) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(diffs_complex(C),X)) ).

% diffs_minus
tff(fact_8475_diffs__minus,axiom,
    ! [C: fun_nat_rat,X: nat] : aa_nat_rat(diffs_rat(aTP_Lamm_ty(C)),X) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(diffs_rat(C),X)) ).

% diffs_minus
tff(fact_8476_diffs__minus,axiom,
    ! [C: fun_nat_Code_integer,X: nat] : aa_nat_Code_integer(diffs_Code_integer(aTP_Lamm_tz(C)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(diffs_Code_integer(C),X)) ).

% diffs_minus
tff(fact_8477_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ).

% zero_less_binomial
tff(fact_8478_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_8479_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8480_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ) ).

% choose_mult
tff(fact_8481_binomial__absorb__comp,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_8482_arccos__le__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_8483_arccos__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real) )
     => ( ( aa_real_real(arccos,Xb) = aa_real_real(arccos,Y) )
      <=> ( Xb = Y ) ) ) ).

% arccos_eq_iff
tff(fact_8484_arccos__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_8485_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_8486_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xb)) ) ) ) ).

% arcsin_minus
tff(fact_8487_arcsin__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( ( aa_real_real(arcsin,Xb) = aa_real_real(arcsin,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_8488_arcsin__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_8489_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_8490_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_8491_diffs__def,axiom,
    ! [C: fun_nat_rat,X: nat] : aa_nat_rat(diffs_rat(C),X) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X))),aa_nat_rat(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8492_diffs__def,axiom,
    ! [C: fun_nat_int,X: nat] : aa_nat_int(diffs_int(C),X) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X))),aa_nat_int(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8493_diffs__def,axiom,
    ! [C: fun_nat_real,X: nat] : aa_nat_real(diffs_real(C),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X))),aa_nat_real(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8494_diffs__def,axiom,
    ! [C: fun_nat_Code_integer,X: nat] : aa_nat_Code_integer(diffs_Code_integer(C),X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,X))),aa_nat_Code_integer(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8495_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8496_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8497_binomial__mono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_8498_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_8499_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_8500_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8501_binomial__antimono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_8502_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_8503_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_8504_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8505_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8506_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_8507_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_8508_termdiff__converges__all,axiom,
    ! [C: fun_nat_complex,Xb: complex] :
      ( ! [X2: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),X2))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_tt(C),Xb)) ) ).

% termdiff_converges_all
tff(fact_8509_termdiff__converges__all,axiom,
    ! [C: fun_nat_real,Xb: real] :
      ( ! [X2: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),X2))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_tv(C),Xb)) ) ).

% termdiff_converges_all
tff(fact_8510_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_8511_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8512_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_8513_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_8514_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_8515_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8516_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_8517_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_8518_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8519_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8520_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8521_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_8522_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_8523_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_8524_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_8525_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_8526_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_8527_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_8528_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = aa_real_real(uminus_uminus_real,Xb) ) ) ) ).

% arccos_cos2
tff(fact_8529_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_8530_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8531_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_8532_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_8533_termdiff__converges,axiom,
    ! [Xb: real,K5: real,C: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),K5)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X2)),K5)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),X2)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ua(Xb),C)) ) ) ).

% termdiff_converges
tff(fact_8534_termdiff__converges,axiom,
    ! [Xb: complex,K5: real,C: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),K5)
     => ( ! [X2: complex] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X2)),K5)
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),X2)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ub(Xb),C)) ) ) ).

% termdiff_converges
tff(fact_8535_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_8536_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8537_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_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_bounded
tff(fact_8538_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_8539_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_8540_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),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,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_8541_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_8542_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_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)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_8543_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_8544_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_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))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_8545_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),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,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_8546_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_8547_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_uc,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8548_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_ud(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8549_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aTP_Lamm_ue(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8550_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_uf(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8551_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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_ug,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8552_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_uh,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8553_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_ui(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8554_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aTP_Lamm_uj(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8555_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_uk(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8556_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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_ul,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8557_monoseq__def,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,M5)),aa_nat_real(X5,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N4)),aa_nat_real(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8558_monoseq__def,axiom,
    ! [X5: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X5)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => ord_less_eq_set_int(aa_nat_set_int(X5,M5),aa_nat_set_int(X5,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => ord_less_eq_set_int(aa_nat_set_int(X5,N4),aa_nat_set_int(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8559_monoseq__def,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,M5)),aa_nat_rat(X5,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N4)),aa_nat_rat(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8560_monoseq__def,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,M5)),aa_nat_num(X5,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N4)),aa_nat_num(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8561_monoseq__def,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,M5)),aa_nat_nat(X5,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N4)),aa_nat_nat(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8562_monoseq__def,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,M5)),aa_nat_int(X5,N4)) )
        | ! [M5: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N4)),aa_nat_int(X5,M5)) ) ) ) ).

% monoseq_def
tff(fact_8563_monoI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N)),aa_nat_real(X5,M)) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI2
tff(fact_8564_monoI2,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => ord_less_eq_set_int(aa_nat_set_int(X5,N),aa_nat_set_int(X5,M)) )
     => topolo3100542954746470799et_int(X5) ) ).

% monoI2
tff(fact_8565_monoI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N)),aa_nat_rat(X5,M)) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI2
tff(fact_8566_monoI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N)),aa_nat_num(X5,M)) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI2
tff(fact_8567_monoI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N)),aa_nat_nat(X5,M)) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI2
tff(fact_8568_monoI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N)),aa_nat_int(X5,M)) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI2
tff(fact_8569_monoI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,M)),aa_nat_real(X5,N)) )
     => topolo6980174941875973593q_real(X5) ) ).

% monoI1
tff(fact_8570_monoI1,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => ord_less_eq_set_int(aa_nat_set_int(X5,M),aa_nat_set_int(X5,N)) )
     => topolo3100542954746470799et_int(X5) ) ).

% monoI1
tff(fact_8571_monoI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,M)),aa_nat_rat(X5,N)) )
     => topolo4267028734544971653eq_rat(X5) ) ).

% monoI1
tff(fact_8572_monoI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,M)),aa_nat_num(X5,N)) )
     => topolo1459490580787246023eq_num(X5) ) ).

% monoI1
tff(fact_8573_monoI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,M)),aa_nat_nat(X5,N)) )
     => topolo4902158794631467389eq_nat(X5) ) ).

% monoI1
tff(fact_8574_monoI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [M: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,M)),aa_nat_int(X5,N)) )
     => topolo4899668324122417113eq_int(X5) ) ).

% monoI1
tff(fact_8575_atMost__iff,axiom,
    ! [I2: real,K: real] :
      ( member_real(I2,set_ord_atMost_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),K) ) ).

% atMost_iff
tff(fact_8576_atMost__iff,axiom,
    ! [I2: set_nat,K: set_nat] :
      ( member_set_nat(I2,set_or4236626031148496127et_nat(K))
    <=> aa_set_nat_o(ord_less_eq_set_nat(I2),K) ) ).

% atMost_iff
tff(fact_8577_atMost__iff,axiom,
    ! [I2: set_int,K: set_int] :
      ( member_set_int(I2,set_or58775011639299419et_int(K))
    <=> ord_less_eq_set_int(I2,K) ) ).

% atMost_iff
tff(fact_8578_atMost__iff,axiom,
    ! [I2: rat,K: rat] :
      ( member_rat(I2,set_ord_atMost_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),K) ) ).

% atMost_iff
tff(fact_8579_atMost__iff,axiom,
    ! [I2: num,K: num] :
      ( member_num(I2,set_ord_atMost_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),K) ) ).

% atMost_iff
tff(fact_8580_atMost__iff,axiom,
    ! [I2: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(set_ord_atMost_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K) ) ).

% atMost_iff
tff(fact_8581_atMost__iff,axiom,
    ! [I2: int,K: int] :
      ( member_int(I2,set_ord_atMost_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K) ) ).

% atMost_iff
tff(fact_8582_atMost__subset__iff,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(Xb),set_or58775011639299419et_int(Y))
    <=> ord_less_eq_set_int(Xb,Y) ) ).

% atMost_subset_iff
tff(fact_8583_atMost__subset__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(Xb),set_ord_atMost_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8584_atMost__subset__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(Xb),set_ord_atMost_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8585_atMost__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8586_atMost__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ord_less_eq_set_int(set_ord_atMost_int(Xb),set_ord_atMost_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8587_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))
    <=> ( ~ ord_less_eq_set_int(L,H)
        | ord_less_eq_set_int(H,H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8588_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H2: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H),set_ord_atMost_rat(H2))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8589_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H2: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(L,H),set_ord_atMost_num(H2))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8590_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_or1269000886237332187st_nat(L,H)),aa_nat_set_nat(set_ord_atMost_nat,H2))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8591_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( ord_less_eq_set_int(set_or1266510415728281911st_int(L,H),set_ord_atMost_int(H2))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8592_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H2: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(L,H),set_ord_atMost_real(H2))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8593_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8594_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8595_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8596_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8597_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_cb(U)) ).

% atMost_def
tff(fact_8598_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_cd(U)) ).

% atMost_def
tff(fact_8599_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_um,U)) ).

% atMost_def
tff(fact_8600_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_un,U)) ).

% atMost_def
tff(fact_8601_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bz,U)) ).

% atMost_def
tff(fact_8602_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_uo,U)) ).

% atMost_def
tff(fact_8603_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_8604_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ ord_less_eq_set_int(set_ord_atMost_int(H),set_or1266510415728281911st_int(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_8605_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ ord_less_eq_set_real(set_ord_atMost_real(H),set_or1222579329274155063t_real(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_8606_finite__nat__iff__bounded__le,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_8607_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(A),set_ord_lessThan_rat(B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8608_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(A),set_ord_lessThan_num(B))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8609_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,A)),aa_nat_set_nat(set_ord_lessThan_nat,B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8610_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(set_ord_atMost_int(A),aa_int_set_int(set_ord_lessThan_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8611_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ord_less_eq_set_real(set_ord_atMost_real(A),set_or5984915006950818249n_real(B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_8612_Iic__subset__Iio__iff,axiom,
    ! [A: $o,B: $o] :
      ( ord_less_eq_set_o(set_ord_atMost_o((A)),aa_o_set_o(set_ord_lessThan_o,(B)))
    <=> aa_o_o(ord_less_o((A)),(B)) ) ).

% Iic_subset_Iio_iff
tff(fact_8613_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_up,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_8614_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ly(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8615_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_lz(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8616_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lo(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8617_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_8618_sum__telescope,axiom,
    ! [F: fun_nat_rat,I2: nat] : groups2906978787729119204at_rat(aTP_Lamm_rf(F),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_8619_sum__telescope,axiom,
    ! [F: fun_nat_int,I2: nat] : groups3539618377306564664at_int(aTP_Lamm_rg(F),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_8620_sum__telescope,axiom,
    ! [F: fun_nat_real,I2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rh(F)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_8621_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,D: fun_nat_complex] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(D),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_complex(C,I5) = aa_nat_complex(D,I5) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8622_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,D: fun_nat_real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(D),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_real(C,I5) = aa_nat_real(D,I5) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8623_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B3: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B3)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_8624_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B3)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_8625_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B3: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B3)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_8626_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_uq(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ur(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_8627_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_us(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ut(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_8628_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uu,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_8629_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uv,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8630_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_uv,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_8631_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,K: nat] :
      ( ! [W2: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8632_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,K: nat] :
      ( ! [W2: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8633_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_complex(C,I5) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_8634_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_real(C,I5) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_8635_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ly(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8636_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_lz(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8637_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_lo(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8638_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lp(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_8639_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8640_sum__up__index__split,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8641_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8642_sum__up__index__split,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_8643_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_uw(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_uy(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_8644_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_uw(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_8645_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vb(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_8646_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vc(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_8647_sum__gp__basic,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8648_sum__gp__basic,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8649_sum__gp__basic,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8650_sum__gp__basic,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_8651_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_vd(C),Nb)))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
          & ( aa_nat_complex(C,I5) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_8652_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_ve(C),Nb)))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
          & ( aa_nat_real(C,I5) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_8653_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,Nb: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => finite3207457112153483333omplex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_vd(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_8654_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,Nb: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_ve(C),Nb))) ) ) ).

% polyfun_roots_finite
tff(fact_8655_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,Nb: nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex )
     => ~ ! [B2: fun_nat_complex] :
            ~ ! [Z4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z4),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(B2),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8656_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,Nb: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat )
     => ~ ! [B2: fun_nat_rat] :
            ~ ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z4),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(B2),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8657_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,Nb: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int )
     => ~ ! [B2: fun_nat_int] :
            ~ ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(B2),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8658_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,Nb: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real )
     => ~ ! [B2: fun_nat_real] :
            ~ ! [Z4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(B2),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_8659_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,Nb: nat,A: complex] :
    ? [B2: fun_nat_complex] :
    ! [Z4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z4),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(B2),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8660_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,Nb: nat,A: rat] :
    ? [B2: fun_nat_rat] :
    ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z4),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(B2),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8661_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,Nb: nat,A: int] :
    ? [B2: fun_nat_int] :
    ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),A)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(B2),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8662_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,Nb: nat,A: real] :
    ? [B2: fun_nat_real] :
    ! [Z4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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,Z4),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(B2),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_8663_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Mb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8664_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(Mb,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8665_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),set_or1269000886237332187st_nat(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Xb),Mb)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8666_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(Mb,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_8667_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vh(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_uy(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_8668_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_vh(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_va(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_8669_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_8670_summable__Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_nz(A))
     => ( summable_real(aTP_Lamm_nz(B))
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vj(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_8671_summable__Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ob(A))
     => ( summable_real(aTP_Lamm_ob(B))
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_vl(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_8672_Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_nz(A))
     => ( summable_real(aTP_Lamm_nz(B))
       => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(A)),suminf_complex(B)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vj(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_8673_Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ob(A))
     => ( summable_real(aTP_Lamm_ob(B))
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(A)),suminf_real(B)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_vl(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_8674_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vm(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_8675_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_mv(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8676_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_mw(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8677_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mx(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8678_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_my(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_8679_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_complex,Nb: nat,B: fun_nat_complex,Xb: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_complex(A,I3) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_vn(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8680_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_rat,Nb: nat,B: fun_nat_rat,Xb: rat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_rat(A,I3) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_vp(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8681_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_int,Nb: nat,B: fun_nat_int,Xb: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_int(A,I3) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_vr(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8682_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_real,Nb: nat,B: fun_nat_real,Xb: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_real(A,I3) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_vs(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_8683_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,Nb: nat,K: complex] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,Nb))
           => ( aa_nat_complex(C,X3) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8684_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,Nb: nat,K: real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_or1269000886237332187st_nat(one_one_nat,Nb))
           => ( aa_nat_real(C,X3) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8685_binomial__ring,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),B)),Nb) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vt(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8686_binomial__ring,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_vu(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8687_binomial__ring,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vv(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8688_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(plus_p5714425477246183910nteger(A,B)),Nb) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_vw(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8689_binomial__ring,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vm(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8690_binomial__ring,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_vx(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_8691_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vy(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vy(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_wa(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_8692_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_wb,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_8693_Cauchy__product__sums,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_nz(A))
     => ( summable_real(aTP_Lamm_nz(B))
       => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vj(A),B)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(A)),suminf_complex(B))) ) ) ).

% Cauchy_product_sums
tff(fact_8694_Cauchy__product__sums,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ob(A))
     => ( summable_real(aTP_Lamm_ob(B))
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_vl(A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(A)),suminf_real(B))) ) ) ).

% Cauchy_product_sums
tff(fact_8695_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wc(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(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,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8696_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_we(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(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,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8697_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wg(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(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,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8698_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wi(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wj(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8699_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P2)
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wk(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wl(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8700_root__polyfun,axiom,
    ! [Nb: nat,Z: int,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wm(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_8701_root__polyfun,axiom,
    ! [Nb: nat,Z: complex,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),Nb) = A )
      <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_wn(Nb),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_8702_root__polyfun,axiom,
    ! [Nb: nat,Z: rat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_rat(power_power_rat(Z),Nb) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_wo(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_8703_root__polyfun,axiom,
    ! [Nb: nat,Z: code_integer,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),Nb) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_wp(Nb),Z),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_8704_root__polyfun,axiom,
    ! [Nb: nat,Z: real,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wq(Nb),Z),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_8705_sum__gp0,axiom,
    ! [Xb: complex,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ).

% sum_gp0
tff(fact_8706_sum__gp0,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp0
tff(fact_8707_sum__gp0,axiom,
    ! [Xb: real,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp0
tff(fact_8708_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_wr,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8709_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_ws(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8710_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups3539618377306564664at_int(aTP_Lamm_wt(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8711_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_wu(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8712_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wv,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8713_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_wx(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_8714_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_rat,Xb: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_wz(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_8715_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_xb(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_8716_polyfun__diff__alt,axiom,
    ! [Nb: nat,A: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_xd(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff_alt
tff(fact_8717_binomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% binomial_r_part_sum
tff(fact_8718_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_xe,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_8719_monoseq__minus,axiom,
    ! [A: fun_nat_int] :
      ( topolo4899668324122417113eq_int(A)
     => topolo4899668324122417113eq_int(aTP_Lamm_tx(A)) ) ).

% monoseq_minus
tff(fact_8720_monoseq__minus,axiom,
    ! [A: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(A)
     => topolo4267028734544971653eq_rat(aTP_Lamm_ty(A)) ) ).

% monoseq_minus
tff(fact_8721_monoseq__minus,axiom,
    ! [A: fun_nat_Code_integer] :
      ( topolo2919662092509805066nteger(A)
     => topolo2919662092509805066nteger(aTP_Lamm_tz(A)) ) ).

% monoseq_minus
tff(fact_8722_monoseq__minus,axiom,
    ! [A: fun_nat_real] :
      ( topolo6980174941875973593q_real(A)
     => topolo6980174941875973593q_real(aTP_Lamm_fy(A)) ) ).

% monoseq_minus
tff(fact_8723_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_xf,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_8724_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_xg(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_8725_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_xh(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8726_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups7501900531339628137nteger(aTP_Lamm_xi(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8727_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_xj,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8728_polyfun__extremal__lemma,axiom,
    ! [E: real,C: fun_nat_complex,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ? [M7: real] :
        ! [Z4: complex] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),aa_complex_real(real_V1022390504157884413omplex,Z4))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z4)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8729_polyfun__extremal__lemma,axiom,
    ! [E: real,C: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ? [M7: real] :
        ! [Z4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),aa_real_real(real_V7735802525324610683m_real,Z4))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Z4)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8730_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_xl(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_8731_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_rat,Xb: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_xn(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_8732_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_vg(A),Y),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_xp(Nb),A),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_8733_polyfun__diff,axiom,
    ! [Nb: nat,A: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(A),Y)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_xr(Nb),A),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ) ).

% polyfun_diff
tff(fact_8734_mono__SucI1,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N)),aa_nat_real(X5,aa_nat_nat(suc,N)))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI1
tff(fact_8735_mono__SucI1,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(X5,N),aa_nat_set_int(X5,aa_nat_nat(suc,N)))
     => topolo3100542954746470799et_int(X5) ) ).

% mono_SucI1
tff(fact_8736_mono__SucI1,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N)),aa_nat_rat(X5,aa_nat_nat(suc,N)))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI1
tff(fact_8737_mono__SucI1,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N)),aa_nat_num(X5,aa_nat_nat(suc,N)))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI1
tff(fact_8738_mono__SucI1,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N)),aa_nat_nat(X5,aa_nat_nat(suc,N)))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI1
tff(fact_8739_mono__SucI1,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N)),aa_nat_int(X5,aa_nat_nat(suc,N)))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI1
tff(fact_8740_mono__SucI2,axiom,
    ! [X5: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N))),aa_nat_real(X5,N))
     => topolo6980174941875973593q_real(X5) ) ).

% mono_SucI2
tff(fact_8741_mono__SucI2,axiom,
    ! [X5: fun_nat_set_int] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(X5,aa_nat_nat(suc,N)),aa_nat_set_int(X5,N))
     => topolo3100542954746470799et_int(X5) ) ).

% mono_SucI2
tff(fact_8742_mono__SucI2,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N))),aa_nat_rat(X5,N))
     => topolo4267028734544971653eq_rat(X5) ) ).

% mono_SucI2
tff(fact_8743_mono__SucI2,axiom,
    ! [X5: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N))),aa_nat_num(X5,N))
     => topolo1459490580787246023eq_num(X5) ) ).

% mono_SucI2
tff(fact_8744_mono__SucI2,axiom,
    ! [X5: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N))),aa_nat_nat(X5,N))
     => topolo4902158794631467389eq_nat(X5) ) ).

% mono_SucI2
tff(fact_8745_mono__SucI2,axiom,
    ! [X5: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N))),aa_nat_int(X5,N))
     => topolo4899668324122417113eq_int(X5) ) ).

% mono_SucI2
tff(fact_8746_monoseq__Suc,axiom,
    ! [X5: fun_nat_real] :
      ( topolo6980174941875973593q_real(X5)
    <=> ( ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,N4)),aa_nat_real(X5,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N4))),aa_nat_real(X5,N4)) ) ) ).

% monoseq_Suc
tff(fact_8747_monoseq__Suc,axiom,
    ! [X5: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X5)
    <=> ( ! [N4: nat] : ord_less_eq_set_int(aa_nat_set_int(X5,N4),aa_nat_set_int(X5,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : ord_less_eq_set_int(aa_nat_set_int(X5,aa_nat_nat(suc,N4)),aa_nat_set_int(X5,N4)) ) ) ).

% monoseq_Suc
tff(fact_8748_monoseq__Suc,axiom,
    ! [X5: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X5)
    <=> ( ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N4)),aa_nat_rat(X5,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N4))),aa_nat_rat(X5,N4)) ) ) ).

% monoseq_Suc
tff(fact_8749_monoseq__Suc,axiom,
    ! [X5: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X5)
    <=> ( ! [N4: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,N4)),aa_nat_num(X5,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N4))),aa_nat_num(X5,N4)) ) ) ).

% monoseq_Suc
tff(fact_8750_monoseq__Suc,axiom,
    ! [X5: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X5)
    <=> ( ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,N4)),aa_nat_nat(X5,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N4))),aa_nat_nat(X5,N4)) ) ) ).

% monoseq_Suc
tff(fact_8751_monoseq__Suc,axiom,
    ! [X5: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X5)
    <=> ( ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,N4)),aa_nat_int(X5,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N4))),aa_nat_int(X5,N4)) ) ) ).

% monoseq_Suc
tff(fact_8752_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_xs,A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Mb)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8753_gbinomial__partial__row__sum,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_xt(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8754_gbinomial__partial__row__sum,axiom,
    ! [A: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xu,A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8755_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Mb))),one_one_complex))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(aa_com8181001469867462503omplex(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))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8756_gbinomial__r__part__sum,axiom,
    ! [Mb: 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,Mb))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8757_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_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,Mb))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(aa_real_fun_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))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8758_pochhammer__double,axiom,
    ! [Z: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,Z),Nb))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8759_pochhammer__double,axiom,
    ! [Z: rat,Nb: 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))),Nb)) = 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))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),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))))),Nb)) ).

% pochhammer_double
tff(fact_8760_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8761_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(aa_nat3829984526574172007omplex(semiri2816024913162550771omplex(aTP_Lamm_xv),Nb),zero_zero_complex) ).

% of_nat_code
tff(fact_8762_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_xw,Nb),zero_zero_rat) ).

% of_nat_code
tff(fact_8763_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(aa_nat_fun_int_int(semiri8420488043553186161ux_int(aTP_Lamm_xx),Nb),zero_zero_int) ).

% of_nat_code
tff(fact_8764_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_xy),Nb),zero_zero_real) ).

% of_nat_code
tff(fact_8765_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_xz),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_8766_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = semiri4055485073559036834nteger(aTP_Lamm_ya,Nb,zero_z3403309356797280102nteger) ).

% of_nat_code
tff(fact_8767_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_xs,R2)),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8768_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_xt(R2),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8769_gchoose__row__sum__weighted,axiom,
    ! [R2: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_xu,R2)),set_or1269000886237332187st_nat(zero_zero_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8770_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_8771_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).

% gbinomial_0(2)
tff(fact_8772_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_8773_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_8774_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_8775_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_8776_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_8777_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_8778_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_8779_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_8780_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_8781_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_8782_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_8783_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_8784_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_8785_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_8786_pochhammer__pos,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8787_pochhammer__pos,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8788_pochhammer__pos,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8789_pochhammer__pos,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8790_pochhammer__neq__0__mono,axiom,
    ! [A: complex,Mb: nat,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Mb) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8791_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8792_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8793_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Mb) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8794_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8795_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8796_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,one_one_complex),Nb) ).

% pochhammer_fact
tff(fact_8797_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Nb) ).

% pochhammer_fact
tff(fact_8798_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,one_one_int),Nb) ).

% pochhammer_fact
tff(fact_8799_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),Nb) ).

% pochhammer_fact
tff(fact_8800_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Nb) ).

% pochhammer_fact
tff(fact_8801_gbinomial__pochhammer,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,A)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer
tff(fact_8802_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(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,A)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_8803_gbinomial__pochhammer,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_8804_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_8805_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_8806_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_8807_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8808_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8809_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_8810_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8811_pochhammer__nonneg,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8812_pochhammer__nonneg,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8813_pochhammer__nonneg,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8814_pochhammer__nonneg,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8815_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_8816_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_8817_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_8818_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_8819_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_8820_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(suc,K)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_8821_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_8822_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_8823_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8824_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8825_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),one_one_complex)),K)) ).

% gbinomial_absorb_comp
tff(fact_8826_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_8827_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_8828_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_8829_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8830_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_8831_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8832_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),aa_nat_nat(suc,Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_8833_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_8834_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_8835_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_8836_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_8837_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_8838_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_8839_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_8840_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_8841_pochhammer__Suc,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(A),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb)),plus_p5714425477246183910nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) ).

% pochhammer_Suc
tff(fact_8842_pochhammer__rec_H,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8843_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8844_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8845_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8846_pochhammer__rec_H,axiom,
    ! [Z: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8847_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8848_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8849_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8850_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8851_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8852_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8853_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8854_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8855_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8856_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8857_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8858_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8859_pochhammer__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8860_pochhammer__product_H,axiom,
    ! [Z: rat,Nb: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8861_pochhammer__product_H,axiom,
    ! [Z: int,Nb: nat,Mb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),Nb)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8862_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,Mb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8863_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8864_pochhammer__product_H,axiom,
    ! [Z: code_integer,Nb: nat,Mb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8865_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8866_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8867_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8868_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8869_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8870_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K)) ).

% Suc_times_gbinomial
tff(fact_8871_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_8872_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% Suc_times_gbinomial
tff(fact_8873_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(suc,K))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_8874_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_8875_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_8876_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8877_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8878_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8879_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z),Mb)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8880_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8881_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8882_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Mb)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(Z,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8883_gbinomial__parallel__sum,axiom,
    ! [A: complex,Nb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_yb,A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),Nb) ).

% gbinomial_parallel_sum
tff(fact_8884_gbinomial__parallel__sum,axiom,
    ! [A: rat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_yc(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),Nb) ).

% gbinomial_parallel_sum
tff(fact_8885_gbinomial__parallel__sum,axiom,
    ! [A: real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yd,A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_real(aa_real_fun_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,Nb))),one_one_real)),Nb) ).

% gbinomial_parallel_sum
tff(fact_8886_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8887_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_8888_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8889_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K)) ).

% gbinomial_factors
tff(fact_8890_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_8891_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% gbinomial_factors
tff(fact_8892_gbinomial__negated__upper,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(semiri8010041392384452111omplex,K)),A)),one_one_complex)),K)) ).

% gbinomial_negated_upper
tff(fact_8893_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_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(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_8894_gbinomial__negated__upper,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_8895_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),Nb)) ).

% gbinomial_index_swap
tff(fact_8896_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : 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(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),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)),Nb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),Nb)) ).

% gbinomial_index_swap
tff(fact_8897_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),Nb)) ).

% gbinomial_index_swap
tff(fact_8898_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R2),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_8899_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_8900_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(aa_int_fun_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(aa_int_fun_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_8901_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_8902_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_8903_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),Nb) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% pochhammer_same
tff(fact_8904_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% pochhammer_same
tff(fact_8905_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% pochhammer_same
tff(fact_8906_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% pochhammer_same
tff(fact_8907_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% pochhammer_same
tff(fact_8908_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_8909_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_8910_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_8911_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(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(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8912_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8913_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8914_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).

% pochhammer_minus'
tff(fact_8915_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_8916_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] : aa_nat_int(aa_int_fun_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(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K)) ).

% pochhammer_minus'
tff(fact_8917_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_8918_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).

% pochhammer_minus'
tff(fact_8919_pochhammer__minus,axiom,
    ! [B: complex,K: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_8920_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_8921_pochhammer__minus,axiom,
    ! [B: int,K: nat] : aa_nat_int(aa_int_fun_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(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(aa_int_fun_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_8922_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_8923_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_8924_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ye,A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Mb)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A),one_one_complex)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_8925_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_yf(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Mb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_8926_gbinomial__sum__lower__neg,axiom,
    ! [A: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yg,A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_8927_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_yh(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8928_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_yi(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8929_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(A,B)),Nb) = groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_yj(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8930_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yk(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_8931_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: complex,Xb: complex,Y: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_yl(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ym(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_8932_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: rat,Xb: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_yn(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_yo(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_8933_gbinomial__partial__sum__poly,axiom,
    ! [Mb: nat,A: real,Xb: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_yp(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_yq(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly
tff(fact_8934_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_yr,K)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8935_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ys(K),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8936_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_yt,K)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8937_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(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_8938_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8939_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8940_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_yu(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_8941_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_yv(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_8942_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_yw(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_8943_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_yx(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_8944_pochhammer__code,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(A),Nb) = $ite(Nb = zero_zero_nat,one_one_Code_integer,set_fo1084959871951514735nteger(aTP_Lamm_yy(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_8945_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_yz(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_8946_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_za,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_8947_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_zb(Mb),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_8948_gbinomial__sum__nat__pow2,axiom,
    ! [Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_zc,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Mb) ).

% gbinomial_sum_nat_pow2
tff(fact_8949_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: complex,Xb: complex,Y: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_yl(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zd(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8950_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: rat,Xb: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_yn(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ze(Mb),A),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8951_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Mb: nat,A: real,Xb: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_yp(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_zf(Mb),A),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8952_fact__double,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_double
tff(fact_8953_fact__double,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_double
tff(fact_8954_fact__double,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_double
tff(fact_8955_gbinomial__code,axiom,
    ! [A: complex,K: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K) = $ite(K = zero_zero_nat,one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_zg(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_8956_gbinomial__code,axiom,
    ! [A: rat,K: nat] :
      aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_zh(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_8957_gbinomial__code,axiom,
    ! [A: real,K: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_zi(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_8958_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,Z),aa_nat_nat(suc,Nb))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = aa_set_nat_complex(groups6464643781859351333omplex(aa_com8181001469867462503omplex(aTP_Lamm_zj,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))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8959_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(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,Nb))) = groups73079841787564623at_rat(aTP_Lamm_zk(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))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8960_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_zl,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))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8961_sin__x__sin__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_zn(Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_x_sin_y
tff(fact_8962_sin__x__sin__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_zp(Xb),Y)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Y))) ).

% sin_x_sin_y
tff(fact_8963_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sx,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_8964_sums__cos__x__plus__y,axiom,
    ! [Xb: real,Y: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_zr(Xb),Y)),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))) ).

% sums_cos_x_plus_y
tff(fact_8965_sums__cos__x__plus__y,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_zt(Xb),Y)),aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))) ).

% sums_cos_x_plus_y
tff(fact_8966_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_8967_mult__scaleR__right,axiom,
    ! [Xb: real,A: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Y)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) ).

% mult_scaleR_right
tff(fact_8968_mult__scaleR__right,axiom,
    ! [Xb: complex,A: real,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Y)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y)) ).

% mult_scaleR_right
tff(fact_8969_mult__scaleR__left,axiom,
    ! [A: real,Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) ).

% mult_scaleR_left
tff(fact_8970_mult__scaleR__left,axiom,
    ! [A: real,Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),Xb)),Y) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y)) ).

% mult_scaleR_left
tff(fact_8971_inverse__mult__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_8972_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% inverse_mult_distrib
tff(fact_8973_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).

% inverse_mult_distrib
tff(fact_8974_inverse__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(inverse_inverse_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_8975_inverse__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_8976_inverse__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,Xb) = one_one_rat )
    <=> ( Xb = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_8977_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_8978_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_8979_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_8980_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) ).

% inverse_divide
tff(fact_8981_inverse__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B),A) ).

% inverse_divide
tff(fact_8982_inverse__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(B),A) ).

% inverse_divide
tff(fact_8983_scaleR__scaleR,axiom,
    ! [A: real,B: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,B),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Xb) ).

% scaleR_scaleR
tff(fact_8984_scaleR__scaleR,axiom,
    ! [A: real,B: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,A),aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,B),Xb)) = aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),Xb) ).

% scaleR_scaleR
tff(fact_8985_prod_Oneutral__const,axiom,
    ! [A2: set_nat] : groups705719431365010083at_int(aTP_Lamm_zu,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8986_prod_Oneutral__const,axiom,
    ! [A2: set_int] : groups1705073143266064639nt_int(aTP_Lamm_zv,A2) = one_one_int ).

% prod.neutral_const
tff(fact_8987_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_8988_tan__periodic__int,axiom,
    ! [Xb: real,I2: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_int
tff(fact_8989_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_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_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8990_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_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_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_8991_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_8992_real__scaleR__def,axiom,
    ! [A: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Xb) ).

% real_scaleR_def
tff(fact_8993_real__sqrt__inverse,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ).

% real_sqrt_inverse
tff(fact_8994_divide__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_8995_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),B)) ).

% complex_scaleR
tff(fact_8996_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_8997_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D3: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D3),E2)
         => ( aa_real_o(P,D3)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_8998_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).

% int_le_real_less
tff(fact_8999_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),Xb) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9000_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).

% int_less_real_le
tff(fact_9001_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I5: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I5)),pi) ) ).

% sin_zero_iff_int2
tff(fact_9002_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_9003_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))) ).

% real_of_int_div2
tff(fact_9004_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_9005_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_9006_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_9007_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),Mb)) ) ) ).

% fact_div_fact
tff(fact_9008_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_9009_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9010_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ? [X3: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_9011_tan__cot,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xb)) ).

% tan_cot
tff(fact_9012_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9013_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9014_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9015_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_9016_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9017_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I5: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I5)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_9018_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I5)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_9019_sinh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% sinh_real_le_iff
tff(fact_9020_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9021_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_9022_divide__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_9023_prod__int__eq,axiom,
    ! [I2: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I2,J)) = groups1705073143266064639nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_9024_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J))) = groups1705073143266064639nt_int(aTP_Lamm_aj,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)))) ).

% prod_int_plus_eq
tff(fact_9025_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9026_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9027_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9028_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(aa_num_num(bitM,Mb),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9029_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xb)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9030_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_9031_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9032_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_9033_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_9034_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_9035_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9036_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_9037_sinh__le__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_le_cosh_real
tff(fact_9038_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_9039_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)) ) ) ).

% OR_lower
tff(fact_9040_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) ) ).

% or_greater_eq
tff(fact_9041_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_nonneg
tff(fact_9042_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9043_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9044_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_ge_1
tff(fact_9045_semiring__norm_I27_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).

% semiring_norm(27)
tff(fact_9046_semiring__norm_I28_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_9047_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9048_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9049_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9050_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_9051_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_9052_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_9053_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_9054_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_9055_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9056_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9057_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xb)) ) ) ).

% cot_gt_zero
tff(fact_9058_tan__cot_H,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).

% tan_cot'
tff(fact_9059_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_9060_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_9061_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_9062_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( 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))))),Xb) = 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,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_9063_complex__i__mult__minus,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Xb)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% complex_i_mult_minus
tff(fact_9064_divide__i,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),imaginary_unit) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xb) ).

% divide_i
tff(fact_9065_i__squared,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_9066_log__le__cancel__iff,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_9067_log__le__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_9068_one__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_9069_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_9070_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_9071_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_9072_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_9073_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_9074_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_9075_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_9076_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_9077_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_9078_or__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_9079_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_9080_or__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_9081_power2__i,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9082_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9083_complex__i__not__numeral,axiom,
    ! [W: num] : imaginary_unit != numera6690914467698888265omplex(W) ).

% complex_i_not_numeral
tff(fact_9084_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9085_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(6)
tff(fact_9086_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9087_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(3)
tff(fact_9088_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),W) = Z )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_9089_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(5)
tff(fact_9090_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(9)
tff(fact_9091_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_i_not_neg_numeral
tff(fact_9092_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(2)
tff(fact_9093_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(A),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_9094_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_9095_log__of__power__eq,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_9096_less__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).

% less_log_of_power
tff(fact_9097_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb)) ).

% or_not_num_neg.simps(8)
tff(fact_9098_log__mult,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_9099_log__divide,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_9100_le__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Mb)) ) ) ).

% le_log_of_power
tff(fact_9101_log__base__pow,axiom,
    ! [A: real,Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_9102_log__nat__power,axiom,
    ! [Xb: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_nat_power
tff(fact_9103_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xb = one )
           => ! [M: num] :
                ( ( Xaa = aa_num_num(bit0,M) )
               => ( Y != aa_num_num(bit1,M) ) ) )
         => ( ( ( Xb = one )
             => ! [M: num] :
                  ( ( Xaa = aa_num_num(bit1,M) )
                 => ( Y != aa_num_num(bit1,M) ) ) )
           => ( ( ? [N: num] : Xb = aa_num_num(bit0,N)
               => ( ( Xaa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa_num_num(bit0,N) )
                   => ! [M: num] :
                        ( ( Xaa = aa_num_num(bit0,M) )
                       => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa_num_num(bit0,N) )
                     => ! [M: num] :
                          ( ( Xaa = aa_num_num(bit1,M) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
                 => ( ( ? [N: num] : Xb = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa_num_num(bit1,N) )
                         => ! [M: num] :
                              ( ( Xaa = aa_num_num(bit0,M) )
                             => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xb = aa_num_num(bit1,N) )
                           => ! [M: num] :
                                ( ( Xaa = aa_num_num(bit1,M) )
                               => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9104_log2__of__power__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ).

% log2_of_power_eq
tff(fact_9105_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_9106_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb) = 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),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_9107_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_9108_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_9109_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_9110_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_9111_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_9112_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9113_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_9114_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9115_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_9116_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( 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))))),Xb) = 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,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_9117_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = 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))) ).

% Arg_minus_ii
tff(fact_9118_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_9119_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9120_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(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,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_9121_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_9122_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_9123_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_9124_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,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,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_9125_cis__minus__pi__half,axiom,
    aa_real_complex(cis,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_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9126_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(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_9127_or__not__num__neg_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Y = one )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [M: num] :
                  ( ( Xaa = aa_num_num(bit0,M) )
                 => ( ( Y = aa_num_num(bit1,M) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit0,M))) ) ) )
           => ( ( ( Xb = one )
               => ! [M: num] :
                    ( ( Xaa = aa_num_num(bit1,M) )
                   => ( ( Y = aa_num_num(bit1,M) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit1,M))) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa_num_num(bit0,N) )
                   => ( ( Xaa = one )
                     => ( ( Y = aa_num_num(bit0,one) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),one)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa_num_num(bit0,N) )
                     => ! [M: num] :
                          ( ( Xaa = aa_num_num(bit0,M) )
                         => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit0,M))) ) ) )
                 => ( ! [N: num] :
                        ( ( Xb = aa_num_num(bit0,N) )
                       => ! [M: num] :
                            ( ( Xaa = aa_num_num(bit1,M) )
                           => ( ( Y = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N),aa_num_num(bit1,M))) ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa_num_num(bit1,N) )
                         => ( ( Xaa = one )
                           => ( ( Y = one )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),one)) ) ) )
                     => ( ! [N: num] :
                            ( ( Xb = aa_num_num(bit1,N) )
                           => ! [M: num] :
                                ( ( Xaa = aa_num_num(bit0,M) )
                               => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit0,M))) ) ) )
                       => ~ ! [N: num] :
                              ( ( Xb = aa_num_num(bit1,N) )
                             => ! [M: num] :
                                  ( ( Xaa = aa_num_num(bit1,M) )
                                 => ( ( Y = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N),aa_num_num(bit1,M))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_9128_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9129_powr__nonneg__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_9130_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_9131_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_9132_powr__le__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_9133_numeral__powr__numeral__real,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_9134_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_9135_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_9136_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9137_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_9138_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9139_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_9140_square__powr__half,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% square_powr_half
tff(fact_9141_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_9142_powr__powr,axiom,
    ! [Xb: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).

% powr_powr
tff(fact_9143_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ).

% powr_ge_pzero
tff(fact_9144_powr__mono2,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mono2
tff(fact_9145_powr__mono,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B)) ) ) ).

% powr_mono
tff(fact_9146_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).

% cis_mult
tff(fact_9147_cis__divide,axiom,
    ! [A: real,B: real] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).

% cis_divide
tff(fact_9148_powr__less__mono2,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_9149_powr__mono2_H,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).

% powr_mono2'
tff(fact_9150_powr__le1,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_9151_powr__mono__both,axiom,
    ! [A: real,B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_9152_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_9153_powr__divide,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_9154_powr__mult,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_9155_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ).

% inverse_powr
tff(fact_9156_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_9157_ln__powr,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_9158_log__powr,axiom,
    ! [Xb: real,B: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ).

% log_powr
tff(fact_9159_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_9160_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb) ) ) ).

% powr_realpow
tff(fact_9161_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_9162_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_9163_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_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_ge
tff(fact_9164_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9165_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_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_ge_diff_one
tff(fact_9166_DeMoivre,axiom,
    ! [A: real,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre
tff(fact_9167_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_9168_le__log__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_9169_log__le__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)) ) ) ) ).

% log_le_iff
tff(fact_9170_le__powr__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_9171_powr__le__iff,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_9172_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_9173_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_9174_ln__powr__bound,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_9175_ln__powr__bound2,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xb)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_9176_add__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = 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)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_9177_log__add__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_9178_minus__log__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) = 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)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_9179_log__minus__eq__powr,axiom,
    ! [B: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9180_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_9181_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_9182_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9183_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,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,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9184_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => bij_betw_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_zw,Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cf,Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9185_summable__complex__of__real,axiom,
    ! [F: fun_nat_real] :
      ( summable_complex(aTP_Lamm_zx(F))
    <=> summable_real(F) ) ).

% summable_complex_of_real
tff(fact_9186_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_9187_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_9188_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9189_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9190_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A3: complex,R3: real] : Z = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A3)) ).

% complex_exp_exists
tff(fact_9191_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_9192_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(complex2(Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_9193_complex__of__real__add__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),Xb)),Y) ).

% complex_of_real_add_Complex
tff(fact_9194_Complex__add__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(complex2(Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),R2)),Y) ).

% Complex_add_complex_of_real
tff(fact_9195_cis__conv__exp,axiom,
    ! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% cis_conv_exp
tff(fact_9196_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_9197_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_9198_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_9199_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A3: real] : Z = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A3))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A3))))) ).

% complex_split_polar
tff(fact_9200_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_9201_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9202_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9203_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_zy(D))) ).

% int_ge_less_than_def
tff(fact_9204_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_zz(D))) ).

% int_ge_less_than2_def
tff(fact_9205_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I3: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I3),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I3),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_9206_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9207_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xb)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xb)) ) ) ).

% of_real_sqrt
tff(fact_9208_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => bij_be1856998921033663316omplex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cf,Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_aaa(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9209_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_aab,Y)) ).

% arctan_def
tff(fact_9210_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_aac,Y)) ).

% arcsin_def
tff(fact_9211_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).

% modulo_int_unfold
tff(fact_9212_real__root__zero,axiom,
    ! [Nb: nat] : aa_real_real(aa_nat_fun_real_real(root,Nb),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9213_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_9214_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9215_root__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_9216_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9217_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_9218_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_9219_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9220_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9221_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9222_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R2))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9223_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R2)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9224_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9225_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9226_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9227_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9228_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9229_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9230_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9231_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9232_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9233_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9234_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_mult
tff(fact_9235_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_minus
tff(fact_9236_real__root__mult__exp,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Xb) = aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_mult_exp
tff(fact_9237_real__root__commute,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_nat_fun_real_real(root,Mb),Xb)) ).

% real_root_commute
tff(fact_9238_real__root__divide,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_divide
tff(fact_9239_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_inverse
tff(fact_9240_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_9241_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_9242_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_9243_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_9244_ln__real__def,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = the_real(aa_real_fun_real_o(aTP_Lamm_aad,Xb)) ).

% ln_real_def
tff(fact_9245_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_9246_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_9247_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9248_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xb) = the_real(aTP_Lamm_bb) ) ) ).

% ln_neg_is_const
tff(fact_9249_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9250_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9251_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9252_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9253_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_9254_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_9255_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9256_real__root__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N2),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9257_real__root__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9258_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_9259_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9260_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_9261_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9262_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9263_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9264_real__root__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N2),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9265_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_aae,Y)) ).

% arccos_def
tff(fact_9266_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) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9267_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9268_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9269_log__base__root,axiom,
    ! [Nb: nat,B: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B),Xb)) ) ) ) ).

% log_base_root
tff(fact_9270_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9271_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9272_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9273_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9274_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_aaf) ).

% pi_half
tff(fact_9275_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_aaf)) ).

% pi_def
tff(fact_9276_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_9277_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_sgn_iff
tff(fact_9278_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_9279_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9280_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xb: real,B: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xb )
     => ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9281_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_9282_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_9283_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,Xb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => ( aa_complex_real(arg,Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_9284_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y3: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y3)),Nb)) = Xb )
             => aa_real_o(P,Y3) ) ) ) ) ).

% split_root
tff(fact_9285_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aa_real_fun_int_o(aTP_Lamm_aag,Xb)) ).

% floor_real_def
tff(fact_9286_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9287_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_9288_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,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(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_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_9289_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9290_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_9291_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_9292_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_9293_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9294_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa_int_nat(nat2,I2) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int) ) ).

% nat_0_iff
tff(fact_9295_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_9296_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_9297_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_9298_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_9299_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_9300_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_9301_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9302_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9303_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_9304_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_9305_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_9306_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9307_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9308_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9309_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9310_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_9311_nat__mask__eq,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% nat_mask_eq
tff(fact_9312_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_9313_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_9314_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
          & aa_nat_o(P,aa_int_nat(nat2,X3)) ) ) ).

% ex_nat
tff(fact_9315_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P,X_12)
    <=> ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_nat_o(P,aa_int_nat(nat2,X3)) ) ) ).

% all_nat
tff(fact_9316_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9317_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9318_unset__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_9319_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_9320_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9321_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_9322_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9323_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_9324_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_9325_int__eq__iff,axiom,
    ! [Mb: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = Z )
    <=> ( ( Mb = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_9326_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_9327_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_9328_int__minus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = 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,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)))) ).

% int_minus
tff(fact_9329_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_9330_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_9331_or__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9332_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_9333_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_9334_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_9335_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(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_9336_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(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_9337_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_9338_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_9339_nat__eq__iff,axiom,
    ! [W: int,Mb: nat] :
      ( ( aa_int_nat(nat2,W) = Mb )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_9340_nat__eq__iff2,axiom,
    ! [Mb: nat,W: int] :
      ( ( Mb = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_9341_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_9342_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ) ).

% le_nat_iff
tff(fact_9343_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_9344_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_9345_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_9346_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z5),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_9347_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9348_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_9349_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_9350_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_9351_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9352_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_9353_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_9354_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_9355_le__nat__floor,axiom,
    ! [Xb: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_9356_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_9357_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_9358_nat__less__iff,axiom,
    ! [W: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),Mb)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).

% nat_less_iff
tff(fact_9359_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9360_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_9361_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_9362_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_9363_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_9364_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_9365_nat__dvd__iff,axiom,
    ! [Z: int,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),Mb)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb)),Mb = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_9366_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_9367_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_9368_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_9369_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).

% even_nat_iff
tff(fact_9370_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_9371_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_9372_powr__int,axiom,
    ! [Xb: real,I2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,I2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I2))))) ) ) ).

% powr_int
tff(fact_9373_floor__rat__def,axiom,
    ! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,Xb) = the_int(aTP_Lamm_aah(Xb)) ).

% floor_rat_def
tff(fact_9374_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_aai(Z))) ).

% Arg_def
tff(fact_9375_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_9376_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),zero_zero_int) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) ).

% concat_bit_of_zero_2
tff(fact_9377_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_9378_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9379_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9380_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S)
           => ! [T4: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T4)
               => ( R2 != aa_rat_rat(plus_plus_rat(S),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9381_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_9382_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9383_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9384_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_9385_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),L))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_9386_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K))) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_9387_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B: int] : aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),B)) = aa_int_fun_int_int(bit_concat_bit(Nb),B) ).

% concat_bit_take_bit_eq
tff(fact_9388_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R2: int,S2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),R2),S2) )
    <=> ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),R2) )
        & ( L = S2 ) ) ) ).

% concat_bit_eq_iff
tff(fact_9389_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Mb),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9390_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9391_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ).

% take_bit_nonnegative
tff(fact_9392_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_9393_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_9394_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9395_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) != zero_zero_int )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9396_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_9397_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_9398_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9399_take__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_9400_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_9401_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_9402_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9403_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_9404_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_9405_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9406_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_9407_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9408_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9409_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = 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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_9410_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_9411_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9412_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_9413_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_9414_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9415_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9416_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_9417_take__bit__numeral__minus__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_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_9418_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_9419_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(minus_minus_rat(Q2),R2) = aa_rat_rat(plus_plus_rat(Q2),aa_rat_rat(uminus_uminus_rat,R2)) ).

% diff_rat_def
tff(fact_9420_num__induct,axiom,
    ! [P: fun_num_o,Xb: num] :
      ( aa_num_o(P,one)
     => ( ! [X2: num] :
            ( aa_num_o(P,X2)
           => aa_num_o(P,aa_num_num(inc,X2)) )
       => aa_num_o(P,Xb) ) ) ).

% num_induct
tff(fact_9421_add__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),Y)) ).

% add_inc
tff(fact_9422_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9423_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit0,Xb)) = aa_num_num(bit1,Xb) ).

% inc.simps(2)
tff(fact_9424_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit1,Xb)) = aa_num_num(bit0,aa_num_num(inc,Xb)) ).

% inc.simps(3)
tff(fact_9425_add__One,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),one) = aa_num_num(inc,Xb) ).

% add_One
tff(fact_9426_inc__BitM__eq,axiom,
    ! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_9427_BitM__inc__eq,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(inc,Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_9428_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xb),aa_num_num(inc,Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Y)),Xb) ).

% mult_inc
tff(fact_9429_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,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,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_9430_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9431_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),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_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_9432_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_9433_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_9434_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9435_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_9436_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9437_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9438_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_9439_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_9440_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9441_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9442_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9443_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9444_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),Nb)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_9445_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),Nb)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        | aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_9446_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_9447_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_9448_AND__upper2,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_9449_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_9450_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)) ) ).

% AND_lower
tff(fact_9451_plus__and__or,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) ).

% plus_and_or
tff(fact_9452_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),one) = Xb ).

% pow.simps(1)
tff(fact_9453_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_9454_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_9455_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K) ) ).

% and_less_eq
tff(fact_9456_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),Nb)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% bit_not_int_iff'
tff(fact_9457_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_9458_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9459_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9460_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9461_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M2)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_9462_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_int_def
tff(fact_9463_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_9464_set__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% set_bit_eq
tff(fact_9465_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_9466_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Nb)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_9467_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
        ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_9468_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( Y = $ite(
            ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xaa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9469_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
            ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_9470_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_9471_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9472_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_9473_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9474_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9475_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9476_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9477_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_9478_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_9479_and__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_9480_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Nb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ) ).

% bit_nat_iff
tff(fact_9481_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb))
     => ( set_or1266510415728281911st_int(Mb,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),set_or1266510415728281911st_int(Mb,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_9482_simp__from__to,axiom,
    ! [I2: int,J: int] :
      set_or1266510415728281911st_int(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),bot_bot_set_int,insert_int(I2,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J))) ).

% simp_from_to
tff(fact_9483_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_9484_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K2: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L3))
           => ( ( ~ ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L3,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9485_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_9486_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9487_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xaa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9488_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_complex(cis,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)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9489_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aaj),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9490_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9491_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9492_set__encode__insert,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Nb),A2)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_9493_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q2),R2) = aa_rat_rat(times_times_rat(Q2),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_9494_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_9495_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_9496_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) ).

% atLeast0_atMost_Suc
tff(fact_9497_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) = set_or1269000886237332187st_nat(Mb,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_9498_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( set_or1269000886237332187st_nat(Mb,aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),set_or1269000886237332187st_nat(Mb,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_9499_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( set_or1269000886237332187st_nat(Mb,Nb) = insert_nat(Mb,set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_9500_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_9501_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_9502_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),pi)) = zero_zero_real )
    <=> member_real(Xb,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9503_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aak),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9504_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9505_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aal),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9506_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_aan(Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9507_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9508_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_aap(Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9509_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),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))),Nb)),Z)) = insert_nat(Nb,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9510_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(sin_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)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9511_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(cos_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)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9512_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aar(Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9513_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aat(Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9514_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9515_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aav(Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9516_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aax(Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9517_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9518_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9519_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9520_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).

% xor_nat_numerals(4)
tff(fact_9521_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_9522_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_9523_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_9524_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K) ).

% Frct_code_post(4)
tff(fact_9525_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9526_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9527_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9528_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_9529_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9530_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9531_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),zero_zero_int),L) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),L) ).

% concat_bit_of_zero_1
tff(fact_9532_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9533_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_9534_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9535_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),Nb)
    <=> ~ ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb)
        <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_9536_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int)) ).

% flip_bit_int_def
tff(fact_9537_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K)) ).

% push_bit_nat_eq
tff(fact_9538_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Y)) ) ) ).

% XOR_lower
tff(fact_9539_set__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9540_flip__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9541_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Mb),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9542_xor__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9543_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9544_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),L)) ).

% concat_bit_eq
tff(fact_9545_concat__bit__def,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Nb),K)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),L)) ).

% concat_bit_def
tff(fact_9546_set__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int)) ).

% set_bit_int_def
tff(fact_9547_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9548_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_int_def
tff(fact_9549_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_minus_one
tff(fact_9550_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Y)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9551_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9552_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9553_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M5)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M5)),aa_nat_real(X5,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9554_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9555_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_9556_and__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9557_or__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9558_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),Nb)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Nb) ) ).

% bit_not_int_iff
tff(fact_9559_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9560_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9561_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9562_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9563_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),one_one_int))) ).

% unset_bit_int_def
tff(fact_9564_xor__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9565_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9566_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_not_iff_int
tff(fact_9567_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9568_and__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(4)
tff(fact_9569_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9570_or__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9571_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),Nb)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9572_numeral__or__not__num__eq,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9573_int__numeral__not__or__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Nb),Mb))) ).

% int_numeral_not_or_num_neg
tff(fact_9574_int__numeral__or__not__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Mb),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9575_and__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9576_and__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(7)
tff(fact_9577_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9578_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9579_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9580_and__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9581_and__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9582_or__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9583_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9584_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9585_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9586_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9587_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9588_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9589_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9590_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9591_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less_eq
tff(fact_9592_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less_eq
tff(fact_9593_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_9594_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(Nb,set_or4665077453230672383an_nat(zero_zero_nat,Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_9595_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9596_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      set_or4665077453230672383an_nat(Mb,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,set_or4665077453230672383an_nat(Mb,Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9597_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9598_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9599_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      set_or4665077453230672383an_nat(Mb,aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(Mb,aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9600_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9601_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_9602_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I3)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aay(A),B)),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),set_or4665077453230672383an_nat(zero_zero_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9603_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_9604_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9605_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
    <=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq
tff(fact_9606_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),D)
     => aa_nat_o(vEBT_VEBT_valid(Ta),D) ) ).

% valid_eq1
tff(fact_9607_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
     => aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq2
tff(fact_9608_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9609_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9610_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_9611_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9612_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = aa_num_real(numeral_numeral_real,V) ).

% complex_Re_numeral
tff(fact_9613_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Re_divide_numeral
tff(fact_9614_sums__Re,axiom,
    ! [X5: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(X5),A)
     => aa_real_o(sums_real(aTP_Lamm_aaz(X5)),aa_complex_real(re,A)) ) ).

% sums_Re
tff(fact_9615_Cauchy__Re,axiom,
    ! [X5: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X5)
     => topolo4055970368930404560y_real(aTP_Lamm_aaz(X5)) ) ).

% Cauchy_Re
tff(fact_9616_complex__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_Re_le_cmod
tff(fact_9617_plus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y)) ).

% plus_complex.simps(1)
tff(fact_9618_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(re,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb)) ).

% scaleR_complex.simps(1)
tff(fact_9619_minus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y)) ).

% minus_complex.simps(1)
tff(fact_9620_summable__Re,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_aaz(F)) ) ).

% summable_Re
tff(fact_9621_abs__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Re_le_cmod
tff(fact_9622_Re__csqrt,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))) ).

% Re_csqrt
tff(fact_9623_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real)
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9624_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,A)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_9625_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
            $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9626_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9627_csqrt__of__real__nonpos,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xb) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9628_Im__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_9629_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_9630_Im__i__times,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z)) = aa_complex_real(re,Z) ).

% Im_i_times
tff(fact_9631_Re__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),Nb) ) ) ).

% Re_power_real
tff(fact_9632_Re__i__times,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% Re_i_times
tff(fact_9633_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Im_divide_numeral
tff(fact_9634_csqrt__of__real__nonneg,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb))
       => ( aa_complex_complex(csqrt,Xb) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,Xb))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_9635_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xb)),zero_zero_real)
        | ( ( aa_complex_real(im,Xb) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(csqrt,Xb)) ) ) ).

% csqrt_minus
tff(fact_9636_sums__Im,axiom,
    ! [X5: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(X5),A)
     => aa_real_o(sums_real(aTP_Lamm_aba(X5)),aa_complex_real(im,A)) ) ).

% sums_Im
tff(fact_9637_Cauchy__Im,axiom,
    ! [X5: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X5)
     => topolo4055970368930404560y_real(aTP_Lamm_aba(X5)) ) ).

% Cauchy_Im
tff(fact_9638_plus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)) ).

% plus_complex.simps(2)
tff(fact_9639_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(im,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb)) ).

% scaleR_complex.simps(2)
tff(fact_9640_minus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)) ).

% minus_complex.simps(2)
tff(fact_9641_sums__complex__iff,axiom,
    ! [F: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(F),Xb)
    <=> ( aa_real_o(sums_real(aTP_Lamm_aaz(F)),aa_complex_real(re,Xb))
        & aa_real_o(sums_real(aTP_Lamm_aba(F)),aa_complex_real(im,Xb)) ) ) ).

% sums_complex_iff
tff(fact_9642_summable__Im,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_aba(F)) ) ).

% summable_Im
tff(fact_9643_abs__Im__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Im_le_cmod
tff(fact_9644_summable__complex__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
    <=> ( summable_real(aTP_Lamm_aaz(F))
        & summable_real(aTP_Lamm_aba(F)) ) ) ).

% summable_complex_iff
tff(fact_9645_times__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_9646_cmod__Re__le__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_real(im,Xb) = aa_complex_real(im,Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_real_real(abs_abs_real,aa_complex_real(re,Y))) ) ) ).

% cmod_Re_le_iff
tff(fact_9647_cmod__Im__le__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_real(re,Xb) = aa_complex_real(re,Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_real_real(abs_abs_real,aa_complex_real(im,Y))) ) ) ).

% cmod_Im_le_iff
tff(fact_9648_times__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_9649_plus__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% plus_complex.code
tff(fact_9650_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xb: complex] : aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,R2),Xb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb))) ).

% scaleR_complex.code
tff(fact_9651_minus__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% minus_complex.code
tff(fact_9652_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).

% csqrt_principal
tff(fact_9653_cmod__le,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))) ).

% cmod_le
tff(fact_9654_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,A)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_9655_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_9656_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_9657_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_9658_times__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_9659_exp__eq__polar,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,Z) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z)))),aa_real_complex(cis,aa_complex_real(im,Z))) ).

% exp_eq_polar
tff(fact_9660_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9661_Im__power2,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),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))),aa_complex_real(re,Xb))),aa_complex_real(im,Xb)) ).

% Im_power2
tff(fact_9662_Re__power2,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9663_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9664_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9665_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9666_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_9667_Re__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)) = 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,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9668_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_9669_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9670_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9671_Im__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y)) = 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,aa_complex_real(im,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9672_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_9673_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9674_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa_complex_complex(invers8013647133539491842omplex,Xb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9675_Complex__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Y) = aa_real_complex(complex2(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,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),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,aa_complex_real(im,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9676_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9677_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,R2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9678_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member_complex(Y,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Y) = Xb )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9679_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member_complex(Y,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( Xb = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),Y) )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9680_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9681_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9682_complex__cnj__mult,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_mult
tff(fact_9683_complex__cnj__power,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(cnj,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cnj,Xb)),Nb) ).

% complex_cnj_power
tff(fact_9684_complex__cnj__add,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_add
tff(fact_9685_complex__cnj__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,numera6690914467698888265omplex(W)) = numera6690914467698888265omplex(W) ).

% complex_cnj_numeral
tff(fact_9686_complex__cnj__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_diff
tff(fact_9687_complex__cnj__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(cnj,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% complex_cnj_neg_numeral
tff(fact_9688_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9689_sums__cnj,axiom,
    ! [F: fun_nat_complex,L: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_abb(F)),aa_complex_complex(cnj,L))
    <=> aa_complex_o(sums_complex(F),L) ) ).

% sums_cnj
tff(fact_9690_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_9691_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_9692_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9693_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_9694_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_gt_0
tff(fact_9695_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_9696_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_ge_0
tff(fact_9697_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_9698_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_gt_0
tff(fact_9699_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_9700_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_ge_0
tff(fact_9701_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9702_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B)))) ) ) ).

% complex_div_gt_0
tff(fact_9703_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9704_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9705_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A),B) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9706_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,W))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9707_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_fa(L),Qr) ).

% divmod_step_integer_def
tff(fact_9708_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ca,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_9709_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_9710_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_9711_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bz,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9712_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_9713_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_9714_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_9715_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9716_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9717_divmod__integer_H__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb)),modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,Mb),aa_num_Code_integer(numera6620942414471956472nteger,Nb))) ).

% divmod_integer'_def
tff(fact_9718_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9719_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_9720_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_9721_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9722_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9723_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_abc),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9724_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_az),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9725_card__less__Suc2,axiom,
    ! [M6: set_nat,I2: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_abd(M6),I2))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_abe(M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_9726_card__less__Suc,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_abd(M6),I2)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_abe(M6),I2))) ) ) ).

% card_less_Suc
tff(fact_9727_card__less,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_abe(M6),I2))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9728_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))))
     => ( A2 = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_set_nat_nat(finite_card_nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9729_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9730_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9731_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_abf,Nb)),Mb) ).

% max_Suc2
tff(fact_9732_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_abg,Nb)),Mb) ).

% max_Suc1
tff(fact_9733_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N2)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9734_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),S3)) ).

% card_sum_le_nat_sum
tff(fact_9735_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_aaa(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9736_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cf,Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9737_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ai),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_9738_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9739_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xaa),Xb)) ).

% plus_integer.abs_eq
tff(fact_9740_times__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xaa),Xb)) ).

% times_integer.abs_eq
tff(fact_9741_less__eq__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xaa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_9742_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : minus_8373710615458151222nteger(aa_int_Code_integer(code_integer_of_int,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xaa),Xb)) ).

% minus_integer.abs_eq
tff(fact_9743_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_9744_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      aa_num_Code_integer(code_integer_of_num,aa_num_num(bit1,Nb)) = $let(
        k: code_integer,
        k:= aa_num_Code_integer(code_integer_of_num,Nb),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9745_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
tff(fact_9746_integer__of__num__triv_I1_J,axiom,
    aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9747_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= aa_num_Code_integer(code_integer_of_num,Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9748_integer__of__num__triv_I2_J,axiom,
    aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9749_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_abh,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9750_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K)) ).

% bit_cut_integer_def
tff(fact_9751_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_abi,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9752_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] : code_int_of_integer(ord_max_Code_integer(K,L)) = aa_int_int(aa_int_fun_int_int(ord_max_int,code_int_of_integer(K)),code_int_of_integer(L)) ).

% int_of_integer_max
tff(fact_9753_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9754_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% plus_integer.rep_eq
tff(fact_9755_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_9756_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(minus_8373710615458151222nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% minus_integer.rep_eq
tff(fact_9757_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9758_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_9759_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_abj(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9760_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_abk,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9761_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,code_nat_of_integer(K)) = ord_max_Code_integer(zero_z3403309356797280102nteger,K) ).

% of_nat_of_integer
tff(fact_9762_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9763_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_9764_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9765_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ai),Nat) ).

% pred_def
tff(fact_9766_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_abl(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_abm(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9767_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_abn(Nb),K))) ).

% binomial_def
tff(fact_9768_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9769_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      nat_prod_decode_aux(K,Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Mb)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_9770_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9771_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9772_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9773_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9774_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9775_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9776_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9777_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Mb),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Nb),K)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),K)) ).

% drop_bit_push_bit_int
tff(fact_9778_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_int_def
tff(fact_9779_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9780_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9781_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9782_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9783_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ).

% fst_divmod_nat
tff(fact_9784_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Nb),K)) ).

% drop_bit_nat_eq
tff(fact_9785_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9786_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb)))) ).

% one_mod_minus_numeral
tff(fact_9787_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_mod_numeral
tff(fact_9788_minus__numeral__mod__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_9789_numeral__mod__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(Mb,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_9790_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      aa_int_int(aa_int_fun_int_int(adjust_mod,L),R2) = $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2)) ).

% Divides.adjust_mod_def
tff(fact_9791_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)))))) ).

% bezw.simps
tff(fact_9792_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_9793_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_9794_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xb,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_9795_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9796_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S3)))
          & ! [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N7),aa_set_nat_nat(finite_card_nat,S3))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N7)),S3) ) ) ) ).

% finite_enumerate
tff(fact_9797_gcd__neg__numeral__1__int,axiom,
    ! [Nb: num,Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),Xb) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,Nb)),Xb) ).

% gcd_neg_numeral_1_int
tff(fact_9798_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,Nb: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_num_int(numeral_numeral_int,Nb)) ).

% gcd_neg_numeral_2_int
tff(fact_9799_gcd__ge__0__int,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ).

% gcd_ge_0_int
tff(fact_9800_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y) ).

% bezout_int
tff(fact_9801_gcd__mult__distrib__int,axiom,
    ! [K: int,Mb: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9802_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) ) ).

% gcd_le2_int
tff(fact_9803_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A) ) ).

% gcd_le1_int
tff(fact_9804_gcd__cases__int,axiom,
    ! [Xb: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9805_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9806_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Mb: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
         => aa_nat_o(P,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
           => ( ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I)
                 => aa_nat_o(P,I) )
             => aa_nat_o(P,K2) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_9807_gcd__1__nat,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9808_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9809_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb))
    <=> ( ( Mb != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9810_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9811_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9812_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9813_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9814_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9815_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X2: nat,Y4: nat] : 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),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9816_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X2: nat,Y4: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9817_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9818_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9819_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_9820_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_num_int(neg_numeral_sub_int(Nb),one))) ).

% xor_minus_numerals(2)
tff(fact_9821_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(neg_numeral_sub_int(Nb),one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9822_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_9823_sub__BitM__One__eq,axiom,
    ! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).

% sub_BitM_One_eq
tff(fact_9824_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_9825_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_9826_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9827_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bz,aTP_Lamm_ca) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9828_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_abo) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9829_Gcd__remove0__nat,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M6,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9830_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_abq),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_9831_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X2: nat,Y4: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4)) ).

% eq_Abs_Integ
tff(fact_9832_Gcd__nat__eq__one,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(member_nat(one_one_nat),N2)
     => ( aa_set_nat_nat(gcd_Gcd_nat,N2) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9833_nat_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xb) ).

% nat.abs_eq
tff(fact_9834_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_9835_int__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),zero_zero_nat)) ).

% int_def
tff(fact_9836_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_abr),Xb)) ).

% uminus_int.abs_eq
tff(fact_9837_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_9838_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_abt),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_9839_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_abv),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_9840_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_abx),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_9841_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_abz),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_9842_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K5)) ).

% Gcd_int_greater_eq_0
tff(fact_9843_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_abv),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9844_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_abt),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_9845_nat_Orep__eq,axiom,
    ! [Xb: int] : aa_int_nat(nat2,Xb) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_9846_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_aca) ).

% prod_encode_def
tff(fact_9847_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_abr)) ).

% uminus_int_def
tff(fact_9848_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_2
tff(fact_9849_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ).

% le_prod_encode_1
tff(fact_9850_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_9851_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_abq)) ).

% times_int_def
tff(fact_9852_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_abz)) ).

% minus_int_def
tff(fact_9853_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_abx)) ).

% plus_int_def
tff(fact_9854_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_acb)) ).

% pred_nat_def
tff(fact_9855_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_9856_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_9857_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9858_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9859_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_9860_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_9861_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9862_pow_Osimps_I3_J,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),aa_num_num(bit1,Y)) = aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,Xb),Y))),Xb) ).

% pow.simps(3)
tff(fact_9863_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9864_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,Nb))) ).

% sqr.simps(2)
tff(fact_9865_sqr__conv__mult,axiom,
    ! [Xb: num] : aa_num_num(sqr,Xb) = aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Xb) ).

% sqr_conv_mult
tff(fact_9866_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,Xb),Y)) ).

% pow.simps(2)
tff(fact_9867_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_9868_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9869_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9870_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(fract(A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9871_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acc,C),set_or4665077453230672383an_nat(Xb,Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9872_bij__betw__Suc,axiom,
    ! [M6: set_nat,N2: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N2)
    <=> ( image_nat_nat(suc,M6) = N2 ) ) ).

% bij_betw_Suc
tff(fact_9873_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I2,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I2),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9874_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I2,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I2),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9875_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_rat_rat(times_times_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(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_rat
tff(fact_9876_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_rat_rat(divide_divide_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(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)) ).

% divide_rat
tff(fact_9877_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_9878_sgn__rat,axiom,
    ! [A: int,B: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(fract(A),B)) = aa_int_rat(ring_1_of_int_rat,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_rat
tff(fact_9879_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ).

% less_rat
tff(fact_9880_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(plus_plus_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(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,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% add_rat
tff(fact_9881_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_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))) ) ) ) ).

% le_rat
tff(fact_9882_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(minus_minus_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,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% diff_rat
tff(fact_9883_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A2)) ).

% zero_notin_Suc_image
tff(fact_9884_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(fract(A),B) = aa_int_rat(fract(C),D) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ).

% eq_rat(1)
tff(fact_9885_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(fract(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_rat(fract(A),B) ) ) ).

% mult_rat_cancel
tff(fact_9886_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_9887_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa_num_rat(numeral_numeral_rat,K) = aa_int_rat(fract(aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% rat_number_expand(3)
tff(fact_9888_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_num_int(numeral_numeral_int,W)),one_one_int) = aa_num_rat(numeral_numeral_rat,W) ).

% rat_number_collapse(3)
tff(fact_9889_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,Nb) ).

% image_Suc_lessThan
tff(fact_9890_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9891_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9892_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9893_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9894_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9895_Fract__add__one,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,Mb),Nb)),Nb) = aa_rat_rat(plus_plus_rat(aa_int_rat(fract(Mb),Nb)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9896_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9897_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9898_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% Fract_le_one_iff
tff(fact_9899_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_9900_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% rat_number_expand(5)
tff(fact_9901_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) ).

% rat_number_collapse(4)
tff(fact_9902_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_9903_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_9904_Gcd__int__eq,axiom,
    ! [N2: set_nat] : gcd_Gcd_int(image_nat_int(semiri1314217659103216013at_int,N2)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N2)) ).

% Gcd_int_eq
tff(fact_9905_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] : aa_set_nat_nat(gcd_Gcd_nat,image_int_nat(aTP_Lamm_acd,K5)) = aa_int_nat(nat2,gcd_Gcd_int(K5)) ).

% Gcd_nat_abs_eq
tff(fact_9906_finite__int__iff__bounded,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
    <=> ? [K3: int] : ord_less_eq_set_int(image_int_int(abs_abs_int,S3),aa_int_set_int(set_ord_lessThan_int,K3)) ) ).

% finite_int_iff_bounded
tff(fact_9907_finite__int__iff__bounded__le,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
    <=> ? [K3: int] : ord_less_eq_set_int(image_int_int(abs_abs_int,S3),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_9908_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_9909_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_ace,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_9910_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(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_9911_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_9912_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_9913_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_9914_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acf,Nb),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Nb) ) ) ).

% range_mod
tff(fact_9915_suminf__eq__SUP__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I3))
       => ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_acg(X5),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9916_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_9917_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9918_range__mult,axiom,
    ! [A: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9919_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9920_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_ach,Nb)),Xb)) ).

% root_def
tff(fact_9921_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_9922_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_9923_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_aci(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9924_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_9925_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_9926_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9927_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X6: $o,X7: $o,X8: $o] : size_size_char(aa_o_char(char2((X1),(X22),(X32),(X42),(X52),(X6),(X7)),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_9928_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% nat_of_char_less_256
tff(fact_9929_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9930_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B72))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((B62)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((B52)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((B42)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((B32)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((B22)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((B1)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_9931_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = produc4188289175737317920o_char(aTP_Lamm_acq,code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_9932_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_9933_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_9934_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_9935_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9936_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9937_list__encode_Oelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( ( ( Xb = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X2: nat,Xs2: list_nat] :
              ( ( Xb = cons_nat(X2,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9938_upto__aux__rec,axiom,
    ! [I2: int,J: int,Js: list_int] :
      upto_aux(I2,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),Js,upto_aux(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9939_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9940_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_9941_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_9942_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( set_or4665077453230672383an_nat(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I2,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9943_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat] : nat_list_encode(cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_9944_list__encode_Opelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),Xb)
       => ( ( ( Xb = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X2: nat,Xs2: list_nat] :
                ( ( Xb = cons_nat(X2,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X2,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9945_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J))
     => ( upto(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9946_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9947_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil
tff(fact_9948_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int = upto(I2,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil2
tff(fact_9949_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2)
     => ( upto(I2,J) = nil_int ) ) ).

% upto_empty
tff(fact_9950_upto__single,axiom,
    ! [I2: int] : upto(I2,I2) = cons_int(I2,nil_int) ).

% upto_single
tff(fact_9951_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I2,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9952_length__upto,axiom,
    ! [I2: int,J: int] : size_size_list_int(upto(I2,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I2)),one_one_int)) ).

% length_upto
tff(fact_9953_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9954_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9955_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9956_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9957_atLeastAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or1266510415728281911st_int(I2,J) = set_int2(upto(I2,J)) ).

% atLeastAtMost_upto
tff(fact_9958_distinct__upto,axiom,
    ! [I2: int,J: int] : aa_list_int_o(distinct_int,upto(I2,J)) ).

% distinct_upto
tff(fact_9959_upto__code,axiom,
    ! [I2: int,J: int] : upto(I2,J) = upto_aux(I2,J,nil_int) ).

% upto_code
tff(fact_9960_upto__aux__def,axiom,
    ! [I2: int,J: int,Js: list_int] : upto_aux(I2,J,Js) = append_int(upto(I2,J),Js) ).

% upto_aux_def
tff(fact_9961_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I2,K) = append_int(upto(I2,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9962_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I2,K) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9963_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or4662586982721622107an_int(I2,J) = set_int2(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9964_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or6656581121297822940st_int(I2,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9965_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9966_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      upto(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9967_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( upto(I2,J) = cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9968_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( upto(I2,J) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9969_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or5832277885323065728an_int(I2,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9970_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I2,K) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9971_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xb != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D2 = 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,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D2 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D2,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9972_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D)
             => ( aa_real_real(F,Xb) = aa_real_real(F,Y4) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9973_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9974_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9975_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9976_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9977_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Xb),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_acr(F),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_9978_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,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)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9979_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9980_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9981_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_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(minus_minus_real,B),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9982_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9983_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D3)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9984_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9985_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9986_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9987_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => ? [Y2: real] :
                  ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9988_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ( ! [X2: real] :
            ( member_real(X2,set_or1222579329274155063t_real(A,B))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X2)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9989_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F3,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),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(F3,Z3)) ) ) ) ) ).

% MVT2
tff(fact_9990_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9991_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xb)),aa_real_real(F,Y4)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9992_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y4)),aa_real_real(F,Xb)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9993_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_acs,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S2)) ).

% DERIV_pow
tff(fact_9994_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_act(G),Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9995_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_acu,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_9996_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F3: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_jg(F),N),aa_nat_real(aa_real_fun_nat_real(F3,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X2: real] :
            ( member_real(X2,set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X2)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F3,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X2: real,Y4: real] :
                    ( member_real(X2,set_or1633881224788618240n_real(A,B))
                   => ( member_real(Y4,set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X2),N)),aa_nat_real(aa_real_fun_nat_real(F,Y4),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X2),Y4)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_qo(F),suminf_real(aa_real_fun_nat_real(F3,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9997_DERIV__log,axiom,
    ! [Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => 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)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9998_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_acv(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,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9999_DERIV__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xb,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_acw(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(F,Xb))),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,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_10000_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_10001_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_10002_DERIV__arctan,axiom,
    ! [Xb: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).

% DERIV_arctan
tff(fact_10003_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,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,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_10004_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D2: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
           => ( D2 = 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,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D2,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_10005_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_10006_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( member_real(X2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_acx(F),X2)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_acz(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_acx(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_10007_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_10008_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_10009_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_10010_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ada(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_10011_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ada(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_10012_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_10013_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ss(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_10014_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ss(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_10015_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ss(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_10016_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xb != zero_zero_real )
         => ( ! [M: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ada(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_10017_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ada(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_10018_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
                 => ( ( Xb != C )
                   => ? [T4: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb) ) )
                        & ( aa_real_real(F,Xb) = 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_adb(Diff),C),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_10019_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),B)
                  & ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_adc(Diff),B),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_10020_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C)
                  & ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_adc(Diff),A),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),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,Nb),T4)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_10021_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B3: real] :
      ( ! [M: nat,T4: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M2: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ade(Nb),Diff),B3),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5)),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_adf(Diff),M2),T5)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_10022_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_adg,suminf_real(aa_real_fun_nat_real(aTP_Lamm_adh,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_10023_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_10024_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_adi(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_10025_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_10026_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).

% take_bit_num_simps(1)
tff(fact_10027_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_10028_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_10029_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_adj,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_10030_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_10031_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_adj,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_10032_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb))) ).

% take_bit_num_simps(7)
tff(fact_10033_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_10034_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_10035_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_10036_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_adk(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_10037_and__not__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(4)
tff(fact_10038_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_10039_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_10040_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_10041_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_adl,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_10042_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_adm(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_10043_and__not__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(7)
tff(fact_10044_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(Mb,Nb) = some_num(Q2) )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_10045_and__not__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_adn,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_10046_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_10047_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_10048_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_10049_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_10050_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_adr,product_Pair_nat_num(Nb,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_10051_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ? [L6: real,M7: real] :
            ( ! [X: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_real_real(F,X))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),M7) ) )
            & ! [Y2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),Y2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),M7) )
               => ? [X2: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
                    & ( aa_real_real(F,X2) = Y2 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_10052_isCont__real__sqrt,axiom,
    ! [Xb: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_10053_isCont__real__root,axiom,
    ! [Xb: real,Nb: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),aa_nat_fun_real_real(root,Nb)) ).

% isCont_real_root
tff(fact_10054_isCont__inverse__function2,axiom,
    ! [A: real,Xb: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B)
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
               => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_10055_LIM__less__bound,axiom,
    ! [B: real,Xb: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),Xb)
     => ( ! [X2: real] :
            ( member_real(X2,set_or1633881224788618240n_real(B,Xb))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_10056_isCont__inverse__function,axiom,
    ! [D: real,Xb: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),Xb))),D)
           => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),Xb))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_10057_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
           => ( ! [Z3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F3,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(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(F3,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10058_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X: real] :
                ( ( ( X != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_10059_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] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X: real] :
                ( ( ( X != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X))),R3) )
               => ( aa_real_real(F,X) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_10060_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X: real] :
                ( ( ( X != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_10061_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_ads,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_10062_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_adt(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_10063_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_adt(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_10064_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_10065_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10066_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_10067_monoseq__convergent,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( topolo6980174941875973593q_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X5,I3))),B3)
       => ~ ! [L6: real] : ~ filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_10068_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_adu,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_10069_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hr(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10070_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: real] :
        ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X5,N)) )
     => filterlim_nat_real(aTP_Lamm_adv(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10071_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_adw,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10072_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adx,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10073_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_ady,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10074_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_adz,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10075_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N7)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10076_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10077_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aea(Xb),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10078_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_10079_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_10080_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aeb,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10081_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aec,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10082_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aed,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_10083_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aee,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10084_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_adt(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_10085_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_adt(A)) ) ) ) ).

% summable
tff(fact_10086_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_aef(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_aeg(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10087_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_aeh(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_aeg(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10088_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_aei(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_adt(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10089_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_nu,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10090_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_aei(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_adt(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10091_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),suminf_real(aTP_Lamm_adt(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10092_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))),L3)
              & filterlim_nat_real(aTP_Lamm_aei(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_aej(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10093_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_aej(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_adt(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10094_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_adt(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10095_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_aej(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_adt(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10096_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C4: real] :
        ! [X3: real] : aa_real_real(F,X3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X3),C4) ) ).

% real_bounded_linear
tff(fact_10097_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aek,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10098_dist__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Xb),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).

% dist_real_def
tff(fact_10099_dist__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V3694042436643373181omplex(Xb),Y) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)) ).

% dist_complex_def
tff(fact_10100_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_10101_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_10102_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_10103_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_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)))),set_or5849166863359141190n_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))))))) ).

% filterlim_tan_at_right
tff(fact_10104_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_10105_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_10106_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_10107_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_10108_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_10109_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
         => ? [Y2: real] :
              ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_10110_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ael(Nb),F),at_bot_real,F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10111_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_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))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10112_Gcd__eq__Max,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( ( M6 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_aem,M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_10113_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eu,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_10114_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_10115_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_10116_ln__at__top,axiom,
    filterlim_real_real(ln_ln_real,at_top_real,at_top_real) ).

% ln_at_top
tff(fact_10117_exp__at__top,axiom,
    filterlim_real_real(exp_real,at_top_real,at_top_real) ).

% exp_at_top
tff(fact_10118_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_10119_filterlim__uminus__at__top__at__bot,axiom,
    filterlim_real_real(uminus_uminus_real,at_top_real,at_bot_real) ).

% filterlim_uminus_at_top_at_bot
tff(fact_10120_filterlim__uminus__at__bot__at__top,axiom,
    filterlim_real_real(uminus_uminus_real,at_bot_real,at_top_real) ).

% filterlim_uminus_at_bot_at_top
tff(fact_10121_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S3)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S3))) ) ).

% card_le_Suc_Max
tff(fact_10122_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aen(Mb),Nb)))) ).

% divide_nat_def
tff(fact_10123_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aeo(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_10124_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_aep,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_10125_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_10126_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_10127_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_aeq,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10128_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aer,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10129_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X2)
         => ? [Y2: real] :
              ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10130_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10131_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10132_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ael(Nb),F),at_top_real,F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10133_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_aes(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10134_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_aet(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10135_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ew,Nb))) = aa_int_int(abs_abs_int,Nb) ) ) ).

% Max_divisors_self_int
tff(fact_10136_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_aet(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_10137_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_az,at_top_nat) ).

% eventually_False_sequentially
tff(fact_10138_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_10139_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F4) ) ).

% le_sequentially
tff(fact_10140_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N8: nat] :
        ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
         => aa_nat_o(P,N4) ) ) ).

% eventually_sequentially
tff(fact_10141_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X2)
         => aa_nat_o(P,X2) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10142_open__bool__def,axiom,
    topolo9180104560040979295open_o = topolo4667128019001906403logy_o(sup_sup_set_set_o(image_o_set_o(set_ord_lessThan_o,top_top_set_o),image_o_set_o(set_or6416164934427428222Than_o,top_top_set_o))) ).

% open_bool_def
tff(fact_10143_open__int__def,axiom,
    topolo4325760605701065253en_int = topolo1611008123915946401gy_int(sup_sup_set_set_int(image_int_set_int(set_ord_lessThan_int,top_top_set_int),image_int_set_int(set_or1207661135979820486an_int,top_top_set_int))) ).

% open_int_def
tff(fact_10144_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),Nb) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_aeu(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_10145_eventually__at__right__to__0,axiom,
    ! [P: fun_real_o,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_aev(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10146_eventually__at__top__to__right,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_aew(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_10147_eventually__at__right__to__top,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_aew(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_10148_eventually__at__left__to__right,axiom,
    ! [P: fun_real_o,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
    <=> eventually_real(aTP_Lamm_aex(P),topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,A),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,A)))) ) ).

% eventually_at_left_to_right
tff(fact_10149_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_aey(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_10150_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_aey(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_10151_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F3),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_10152_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F3),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_10153_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_10154_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F3),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_10155_lhopital,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aTP_Lamm_afb(G),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),F4,topolo2177554685111907308n_real(Xb,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),F4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10156_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_10157_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_afb(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),topolo2815343760600316023s_real(Xb),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G),F),topolo2815343760600316023s_real(Xb),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_10158_open__nat__def,axiom,
    topolo4328251076210115529en_nat = topolo1613498594424996677gy_nat(sup_sup_set_set_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat),image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat))) ).

% open_nat_def
tff(fact_10159_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_10160_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_10161_lhopital__right,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aTP_Lamm_afb(G),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),F4,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),F4,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_10162_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_afb(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F0),F3),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),F4,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F0),G0),F4,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_10163_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_10164_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G),F),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10165_lhopital__left,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aTP_Lamm_afb(G),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),F4,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(F),G),F4,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10166_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( eventually_real(aTP_Lamm_afb(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(F),F3),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_aez(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10167_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_10168_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_10169_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_10170_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A: real,B: real] :
      ( ord_less_eq_set_real(image_nat_real(F,top_top_set_nat),set_or1222579329274155063t_real(A,B))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_10171_Bseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xb),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10172_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X3: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                            & ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_10173_decseq__bounded,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),aa_nat_real(X5,I3))
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_10174_decseq__convergent,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),aa_nat_real(X5,I3))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_nat_real(X5,I)) ) ) ) ).

% decseq_convergent
tff(fact_10175_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya)),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Dega),n),
              ( ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X3),n) )
              & aa_nat_o(vEBT_VEBT_valid(Summarya),m2)
              & ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X9)
                    & ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_10176_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_10177_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X3: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                              & ! [X3: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_10178_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_10179_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_10180_Sup__int__def,axiom,
    ! [X5: set_int] : complete_Sup_Sup_int(X5) = the_int(aTP_Lamm_afe(X5)) ).

% Sup_int_def
tff(fact_10181_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_10182_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ( ! [X2: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) )
         => ( ! [X2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G) )
           => ( ! [X2: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X2,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))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(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_10183_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ? [L3: real,Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),B)
              & has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z3,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_10184_continuous__on__arsinh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => topolo5044208981011980120l_real(A2,aTP_Lamm_aff(F)) ) ).

% continuous_on_arsinh'
tff(fact_10185_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X2)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_afg(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_10186_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ? [C3: real,D3: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,B)) = set_or1222579329274155063t_real(C3,D3) )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C3),D3) ) ) ) ).

% continuous_image_closed_interval
tff(fact_10187_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A2,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_10188_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X2: real] :
            ( member_real(X2,A2)
           => member_real(aa_real_real(F,X2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_afh(F)) ) ) ).

% continuous_on_artanh'
tff(fact_10189_mvt,axiom,
    ! [A: real,B: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi),B)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F3,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_10190_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A2,artanh_real) ) ).

% continuous_on_artanh
tff(fact_10191_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
             => ( aa_real_real(F,Xb) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_10192_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_afj,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_10193_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_afl,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_10194_open__complex__def,axiom,
    ! [U3: set_complex] :
      ( topolo4110288021797289639omplex(U3)
    <=> ! [X3: complex] :
          ( member_complex(X3,U3)
         => eventu5826381225784669381omplex(produc6771430404735790350plex_o(aa_com7221863818706917250plex_o(aTP_Lamm_afm(U3),X3)),topolo896644834953643431omplex) ) ) ).

% open_complex_def
tff(fact_10195_open__real__def,axiom,
    ! [U3: set_real] :
      ( topolo4860482606490270245n_real(U3)
    <=> ! [X3: real] :
          ( member_real(X3,U3)
         => eventu3244425730907250241l_real(produc5414030515140494994real_o(aa_rea1652550732481586312real_o(aTP_Lamm_afn(U3),X3)),topolo1511823702728130853y_real) ) ) ).

% open_real_def
tff(fact_10196_less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% less_eq
tff(fact_10197_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M5)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N4),M5)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_10198_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_10199_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10200_incseq__bounded,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I3)),B3)
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_10201_incseq__convergent,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I3)),B3)
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I)),L6) ) ) ) ).

% incseq_convergent
tff(fact_10202_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_oc,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10203_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_tn,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10204_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_aaf,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10205_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_sk,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10206_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ads,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10207_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aep,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10208_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_adu,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_7
tff(fact_10209_ATP_Olambda__8,axiom,
    ! [Uu: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_ya,Uu) = plus_p5714425477246183910nteger(Uu,one_one_Code_integer) ).

% ATP.lambda_8
tff(fact_10210_ATP_Olambda__9,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_xv,Uu) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),one_one_complex) ).

% ATP.lambda_9
tff(fact_10211_ATP_Olambda__10,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_xy,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).

% ATP.lambda_10
tff(fact_10212_ATP_Olambda__11,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_xw,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_11
tff(fact_10213_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_xz,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_12
tff(fact_10214_ATP_Olambda__13,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_xx,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_13
tff(fact_10215_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_adw,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_14
tff(fact_10216_ATP_Olambda__15,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_adg,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_nw,Uu)) ).

% ATP.lambda_15
tff(fact_10217_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_aem,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eu,Uu)) ).

% ATP.lambda_16
tff(fact_10218_ATP_Olambda__17,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_afl,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_afk(Uu)))) ).

% ATP.lambda_17
tff(fact_10219_ATP_Olambda__18,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_afj,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_afi(Uu)))) ).

% ATP.lambda_18
tff(fact_10220_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ady,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_19
tff(fact_10221_ATP_Olambda__20,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_adn,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_20
tff(fact_10222_ATP_Olambda__21,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_adj,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_21
tff(fact_10223_ATP_Olambda__22,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aal,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_22
tff(fact_10224_ATP_Olambda__23,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aak,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_23
tff(fact_10225_ATP_Olambda__24,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_km,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_24
tff(fact_10226_ATP_Olambda__25,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_acd,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_25
tff(fact_10227_ATP_Olambda__26,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_adl,Uu) = some_num(one) ).

% ATP.lambda_26
tff(fact_10228_ATP_Olambda__27,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_adq(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_ado(Uua),aTP_Lamm_adp(Uua),Uu) ).

% ATP.lambda_27
tff(fact_10229_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_uk(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_28
tff(fact_10230_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_uh,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_29
tff(fact_10231_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ul,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_30
tff(fact_10232_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_ui(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_31
tff(fact_10233_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aTP_Lamm_uj(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_32
tff(fact_10234_ATP_Olambda__33,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_to(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_33
tff(fact_10235_ATP_Olambda__34,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aaj,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_34
tff(fact_10236_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_uf(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_35
tff(fact_10237_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_uc,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_36
tff(fact_10238_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ug,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_37
tff(fact_10239_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_ud(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_38
tff(fact_10240_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aTP_Lamm_ue(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_39
tff(fact_10241_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_adr,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_adq(Uua),Uu) ).

% ATP.lambda_40
tff(fact_10242_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_ado(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_adj,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_41
tff(fact_10243_ATP_Olambda__42,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_adk(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_adj,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_42
tff(fact_10244_ATP_Olambda__43,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_va(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_uz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_43
tff(fact_10245_ATP_Olambda__44,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uy(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ux(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_44
tff(fact_10246_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_45
tff(fact_10247_ATP_Olambda__46,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_46
tff(fact_10248_ATP_Olambda__47,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_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_47
tff(fact_10249_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_wu(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_48
tff(fact_10250_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_wr,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(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_49
tff(fact_10251_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_50
tff(fact_10252_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ws(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_51
tff(fact_10253_ATP_Olambda__52,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_wt(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(aa_int_fun_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_52
tff(fact_10254_ATP_Olambda__53,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_za,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_53
tff(fact_10255_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_zc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_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(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_54
tff(fact_10256_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_zb(Uu),Uua) = aa_rat_rat(divide_divide_rat(aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_55
tff(fact_10257_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aac,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_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(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_56
tff(fact_10258_ATP_Olambda__57,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aab,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_57
tff(fact_10259_ATP_Olambda__58,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_abh,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_58
tff(fact_10260_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_nt,Uu),Uua) = $let(
        m3: code_integer,
        m3:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),
        $ite(Uua = zero_zero_nat,m3,plus_p5714425477246183910nteger(m3,one_one_Code_integer)) ) ).

% ATP.lambda_59
tff(fact_10261_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_no,Uu),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,m3),one_one_complex)) ) ).

% ATP.lambda_60
tff(fact_10262_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nr,Uu),Uua) = $let(
        m3: real,
        m3:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m3),one_one_real)) ) ).

% ATP.lambda_61
tff(fact_10263_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_np,Uu),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(plus_plus_rat(m3),one_one_rat)) ) ).

% ATP.lambda_62
tff(fact_10264_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ns,Uu),Uua) = $let(
        m3: nat,
        m3:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_63
tff(fact_10265_ATP_Olambda__64,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nq,Uu),Uua) = $let(
        m3: int,
        m3:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_64
tff(fact_10266_ATP_Olambda__65,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_aai(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_65
tff(fact_10267_ATP_Olambda__66,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_aag,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_66
tff(fact_10268_ATP_Olambda__67,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aah(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_67
tff(fact_10269_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nw,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_68
tff(fact_10270_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_69
tff(fact_10271_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adt(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_70
tff(fact_10272_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_xi(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_71
tff(fact_10273_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_xf,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(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_72
tff(fact_10274_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_xj,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_73
tff(fact_10275_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_xg(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_74
tff(fact_10276_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_xh(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_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_75
tff(fact_10277_ATP_Olambda__76,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_yb,Uu),Uua) = aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_76
tff(fact_10278_ATP_Olambda__77,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yd,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_77
tff(fact_10279_ATP_Olambda__78,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_yc(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_78
tff(fact_10280_ATP_Olambda__79,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_xs,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,Uu),Uua)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_79
tff(fact_10281_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_80
tff(fact_10282_ATP_Olambda__81,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_xt(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_81
tff(fact_10283_ATP_Olambda__82,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ye,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,Uu),Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)) ).

% ATP.lambda_82
tff(fact_10284_ATP_Olambda__83,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_83
tff(fact_10285_ATP_Olambda__84,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_yf(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)) ).

% ATP.lambda_84
tff(fact_10286_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_abo,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_85
tff(fact_10287_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_86
tff(fact_10288_ATP_Olambda__87,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_afe(Uu),Uua)
    <=> ( member_int(Uua,Uu)
        & ! [X3: int] :
            ( member_int(X3,Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Uua) ) ) ) ).

% ATP.lambda_87
tff(fact_10289_ATP_Olambda__88,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sy,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_88
tff(fact_10290_ATP_Olambda__89,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aeq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_89
tff(fact_10291_ATP_Olambda__90,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uv,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_90
tff(fact_10292_ATP_Olambda__91,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uu,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_91
tff(fact_10293_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_cf,Uu),Uua)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_92
tff(fact_10294_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_ce(Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_93
tff(fact_10295_ATP_Olambda__94,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aed,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_94
tff(fact_10296_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aek,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_95
tff(fact_10297_ATP_Olambda__96,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_aer,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).

% ATP.lambda_96
tff(fact_10298_ATP_Olambda__97,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_97
tff(fact_10299_ATP_Olambda__98,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aae,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_98
tff(fact_10300_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_my(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_99
tff(fact_10301_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mv(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_100
tff(fact_10302_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mx(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_101
tff(fact_10303_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mw(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_102
tff(fact_10304_ATP_Olambda__103,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mc(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_103
tff(fact_10305_ATP_Olambda__104,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ma(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_104
tff(fact_10306_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mb(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_105
tff(fact_10307_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_kp(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uua)) ).

% ATP.lambda_106
tff(fact_10308_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kr(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_107
tff(fact_10309_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kq(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_108
tff(fact_10310_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_qx(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Uua)) ).

% ATP.lambda_109
tff(fact_10311_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mu(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_110
tff(fact_10312_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ms(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_111
tff(fact_10313_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mt(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_112
tff(fact_10314_ATP_Olambda__113,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mo(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(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_113
tff(fact_10315_ATP_Olambda__114,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mr(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_114
tff(fact_10316_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mp(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_115
tff(fact_10317_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mq(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_116
tff(fact_10318_ATP_Olambda__117,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rh(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_117
tff(fact_10319_ATP_Olambda__118,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_rf(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_118
tff(fact_10320_ATP_Olambda__119,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_rg(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_119
tff(fact_10321_ATP_Olambda__120,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_120
tff(fact_10322_ATP_Olambda__121,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_uq(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_121
tff(fact_10323_ATP_Olambda__122,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_tm(Uu),Uua) = aa_real_complex(complex2(aa_complex_real(Uu,Uua)),zero_zero_real) ).

% ATP.lambda_122
tff(fact_10324_ATP_Olambda__123,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_tl(Uu),Uua) = aa_real_complex(complex2(aa_nat_real(Uu,Uua)),zero_zero_real) ).

% ATP.lambda_123
tff(fact_10325_ATP_Olambda__124,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kb(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_124
tff(fact_10326_ATP_Olambda__125,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kd(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_125
tff(fact_10327_ATP_Olambda__126,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kf(Uu),Uua)
    <=> ( aa_complex_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_126
tff(fact_10328_ATP_Olambda__127,axiom,
    ! [Uu: fun_complex_int,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kg(Uu),Uua)
    <=> ( aa_complex_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_127
tff(fact_10329_ATP_Olambda__128,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_kh(Uu),Uua)
    <=> ( aa_nat_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_128
tff(fact_10330_ATP_Olambda__129,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ki(Uu),Uua)
    <=> ( aa_nat_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_129
tff(fact_10331_ATP_Olambda__130,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( aa_int_o(aTP_Lamm_jz(Uu),Uua)
    <=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).

% ATP.lambda_130
tff(fact_10332_ATP_Olambda__131,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_ka(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_131
tff(fact_10333_ATP_Olambda__132,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_kc(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_132
tff(fact_10334_ATP_Olambda__133,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ke(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_133
tff(fact_10335_ATP_Olambda__134,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aej(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(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_134
tff(fact_10336_ATP_Olambda__135,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aei(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_adt(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_135
tff(fact_10337_ATP_Olambda__136,axiom,
    ! [Uu: code_integer,Uua: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_acq,Uu),(Uua)) = produc4188289175737317920o_char(aTP_Lamm_acp((Uua)),code_bit_cut_integer(Uu)) ).

% ATP.lambda_136
tff(fact_10338_ATP_Olambda__137,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aca,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_137
tff(fact_10339_ATP_Olambda__138,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ts,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_138
tff(fact_10340_ATP_Olambda__139,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_139
tff(fact_10341_ATP_Olambda__140,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_adi(Uu),Uua) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_140
tff(fact_10342_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ach,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_141
tff(fact_10343_ATP_Olambda__142,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sx,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_142
tff(fact_10344_ATP_Olambda__143,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_143
tff(fact_10345_ATP_Olambda__144,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_abi,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_144
tff(fact_10346_ATP_Olambda__145,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_abk,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_145
tff(fact_10347_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_yr,Uu),Uua) = aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_146
tff(fact_10348_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_yt,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_147
tff(fact_10349_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_ys(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_148
tff(fact_10350_ATP_Olambda__149,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aad,Uu),Uua)
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_149
tff(fact_10351_ATP_Olambda__150,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_kx,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_150
tff(fact_10352_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_kw,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_151
tff(fact_10353_ATP_Olambda__152,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kv,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_152
tff(fact_10354_ATP_Olambda__153,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_kl,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_153
tff(fact_10355_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_kk,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_154
tff(fact_10356_ATP_Olambda__155,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kj,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_155
tff(fact_10357_ATP_Olambda__156,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_zj,Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_156
tff(fact_10358_ATP_Olambda__157,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zl,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(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_157
tff(fact_10359_ATP_Olambda__158,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_zk(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_158
tff(fact_10360_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_xe,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_159
tff(fact_10361_ATP_Olambda__160,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aee,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_160
tff(fact_10362_ATP_Olambda__161,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_md,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_161
tff(fact_10363_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aec,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_162
tff(fact_10364_ATP_Olambda__163,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_163
tff(fact_10365_ATP_Olambda__164,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_acg(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_164
tff(fact_10366_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acb,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_165
tff(fact_10367_ATP_Olambda__166,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_cc(Uu),Uua)
    <=> ord_le211207098394363844omplex(Uua,Uu) ) ).

% ATP.lambda_166
tff(fact_10368_ATP_Olambda__167,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_cb(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_167
tff(fact_10369_ATP_Olambda__168,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_cd(Uu),Uua)
    <=> ord_less_eq_set_int(Uua,Uu) ) ).

% ATP.lambda_168
tff(fact_10370_ATP_Olambda__169,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_um,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_169
tff(fact_10371_ATP_Olambda__170,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_un,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_10372_ATP_Olambda__171,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bz,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_10373_ATP_Olambda__172,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_uo,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_10374_ATP_Olambda__173,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acf,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_173
tff(fact_10375_ATP_Olambda__174,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_qh,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_174
tff(fact_10376_ATP_Olambda__175,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_qi((Uu)),(Uua))
    <=> aa_o_o(ord_less_o((Uua)),(Uu)) ) ).

% ATP.lambda_175
tff(fact_10377_ATP_Olambda__176,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_qd(Uu),Uua)
    <=> aa_set_nat_o(ord_less_set_nat(Uua),Uu) ) ).

% ATP.lambda_176
tff(fact_10378_ATP_Olambda__177,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_qe,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_177
tff(fact_10379_ATP_Olambda__178,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_qf,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_178
tff(fact_10380_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ca,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_179
tff(fact_10381_ATP_Olambda__180,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qg,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_10382_ATP_Olambda__181,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_am,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_181
tff(fact_10383_ATP_Olambda__182,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_an(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_182
tff(fact_10384_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_183
tff(fact_10385_ATP_Olambda__184,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ap,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_184
tff(fact_10386_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_185
tff(fact_10387_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_acs,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_186
tff(fact_10388_ATP_Olambda__187,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ace,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_187
tff(fact_10389_ATP_Olambda__188,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_acu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_188
tff(fact_10390_ATP_Olambda__189,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ev,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_10391_ATP_Olambda__190,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_et,Uu),Uua)
    <=> aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Uua),Uu) ) ).

% ATP.lambda_190
tff(fact_10392_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_eu,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_191
tff(fact_10393_ATP_Olambda__192,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ew,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_192
tff(fact_10394_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_abr,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_193
tff(fact_10395_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_up,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_194
tff(fact_10396_ATP_Olambda__195,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_adx,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_195
tff(fact_10397_ATP_Olambda__196,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aa(Uu),Uua)
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_196
tff(fact_10398_ATP_Olambda__197,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_complex(Uua,Uu) ) ).

% ATP.lambda_197
tff(fact_10399_ATP_Olambda__198,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_a(Uu),Uua)
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_198
tff(fact_10400_ATP_Olambda__199,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_199
tff(fact_10401_ATP_Olambda__200,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_200
tff(fact_10402_ATP_Olambda__201,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_201
tff(fact_10403_ATP_Olambda__202,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sj(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_202
tff(fact_10404_ATP_Olambda__203,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_si(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_203
tff(fact_10405_ATP_Olambda__204,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_aew(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_204
tff(fact_10406_ATP_Olambda__205,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_acr(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_205
tff(fact_10407_ATP_Olambda__206,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_aex(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(uminus_uminus_real,Uua)) ) ).

% ATP.lambda_206
tff(fact_10408_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_pk(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_207
tff(fact_10409_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lp(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_208
tff(fact_10410_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aes(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_209
tff(fact_10411_ATP_Olambda__210,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ly(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_210
tff(fact_10412_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_lo(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_211
tff(fact_10413_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lz(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_212
tff(fact_10414_ATP_Olambda__213,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_adp(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_213
tff(fact_10415_ATP_Olambda__214,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_adm(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_214
tff(fact_10416_ATP_Olambda__215,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abz,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aby(Uu),Uua)) ).

% ATP.lambda_215
tff(fact_10417_ATP_Olambda__216,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abx,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abw(Uu),Uua)) ).

% ATP.lambda_216
tff(fact_10418_ATP_Olambda__217,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_abv,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abu(Uu),Uua)) ).

% ATP.lambda_217
tff(fact_10419_ATP_Olambda__218,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_abt,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abs(Uu),Uua)) ).

% ATP.lambda_218
tff(fact_10420_ATP_Olambda__219,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_abq,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abp(Uu),Uua)) ).

% ATP.lambda_219
tff(fact_10421_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_acz(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_acy(Uu),Uua)) ).

% ATP.lambda_220
tff(fact_10422_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_zw,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_221
tff(fact_10423_ATP_Olambda__222,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_afb(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_222
tff(fact_10424_ATP_Olambda__223,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ex,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_223
tff(fact_10425_ATP_Olambda__224,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_zg(Uu),Uua) = aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_224
tff(fact_10426_ATP_Olambda__225,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_zi(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_225
tff(fact_10427_ATP_Olambda__226,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_zh(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_226
tff(fact_10428_ATP_Olambda__227,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_yy(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(Uu,aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_227
tff(fact_10429_ATP_Olambda__228,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_yu(Uu),Uua) = aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_228
tff(fact_10430_ATP_Olambda__229,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_yx(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_229
tff(fact_10431_ATP_Olambda__230,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_yv(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_230
tff(fact_10432_ATP_Olambda__231,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_yz(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_231
tff(fact_10433_ATP_Olambda__232,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_yw(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_232
tff(fact_10434_ATP_Olambda__233,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aeb,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_233
tff(fact_10435_ATP_Olambda__234,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abg,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_234
tff(fact_10436_ATP_Olambda__235,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_abf,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_235
tff(fact_10437_ATP_Olambda__236,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_le(Uu),Uua)
    <=> ~ member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_236
tff(fact_10438_ATP_Olambda__237,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_lf(Uu),Uua)
    <=> ~ member_complex(Uua,Uu) ) ).

% ATP.lambda_237
tff(fact_10439_ATP_Olambda__238,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ld(Uu),Uua)
    <=> ~ member_real(Uua,Uu) ) ).

% ATP.lambda_238
tff(fact_10440_ATP_Olambda__239,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_lg(Uu),Uua)
    <=> ~ member_set_nat(Uua,Uu) ) ).

% ATP.lambda_239
tff(fact_10441_ATP_Olambda__240,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_lh(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_240
tff(fact_10442_ATP_Olambda__241,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_li(Uu),Uua)
    <=> ~ member_int(Uua,Uu) ) ).

% ATP.lambda_241
tff(fact_10443_ATP_Olambda__242,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_oa(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_242
tff(fact_10444_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_nz(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_243
tff(fact_10445_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ob(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_244
tff(fact_10446_ATP_Olambda__245,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_adv(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_245
tff(fact_10447_ATP_Olambda__246,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_nc(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_246
tff(fact_10448_ATP_Olambda__247,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_ne(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_247
tff(fact_10449_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_ng(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_248
tff(fact_10450_ATP_Olambda__249,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nf(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_249
tff(fact_10451_ATP_Olambda__250,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_nd(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_250
tff(fact_10452_ATP_Olambda__251,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_nb(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_251
tff(fact_10453_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_zx(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_252
tff(fact_10454_ATP_Olambda__253,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_fx(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_253
tff(fact_10455_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_tz(Uu),Uua) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_254
tff(fact_10456_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_pl(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_255
tff(fact_10457_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fy(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_256
tff(fact_10458_ATP_Olambda__257,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ty(Uu),Uua) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_257
tff(fact_10459_ATP_Olambda__258,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_tx(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_258
tff(fact_10460_ATP_Olambda__259,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fw(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_259
tff(fact_10461_ATP_Olambda__260,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_sl(Uu),Uua) = aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_260
tff(fact_10462_ATP_Olambda__261,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_sp(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_261
tff(fact_10463_ATP_Olambda__262,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_sm(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_262
tff(fact_10464_ATP_Olambda__263,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_so(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_263
tff(fact_10465_ATP_Olambda__264,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_sn(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_10466_ATP_Olambda__265,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_afh(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_10467_ATP_Olambda__266,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_aff(Uu),Uua) = aa_real_real(arsinh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_10468_ATP_Olambda__267,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_afg(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_10469_ATP_Olambda__268,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ko(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_10470_ATP_Olambda__269,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_kn(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_269
tff(fact_10471_ATP_Olambda__270,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aeh(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_10472_ATP_Olambda__271,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_aaz(Uu),Uua) = aa_complex_real(re,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_271
tff(fact_10473_ATP_Olambda__272,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_aba(Uu),Uua) = aa_complex_real(im,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_272
tff(fact_10474_ATP_Olambda__273,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: complex] : aa_complex_complex(aTP_Lamm_qu(Uu),Uua) = suminf_complex(aa_com8181001469867462503omplex(Uu,Uua)) ).

% ATP.lambda_273
tff(fact_10475_ATP_Olambda__274,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: complex] : aa_complex_real(aTP_Lamm_qn(Uu),Uua) = suminf_real(aa_com20334259897932645t_real(Uu,Uua)) ).

% ATP.lambda_274
tff(fact_10476_ATP_Olambda__275,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: real] : aa_real_complex(aTP_Lamm_qq(Uu),Uua) = suminf_complex(aa_rea1531280491672885477omplex(Uu,Uua)) ).

% ATP.lambda_275
tff(fact_10477_ATP_Olambda__276,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_qo(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_276
tff(fact_10478_ATP_Olambda__277,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat] : aa_nat_complex(aTP_Lamm_qr(Uu),Uua) = suminf_complex(aa_nat6454746962644518025omplex(Uu,Uua)) ).

% ATP.lambda_277
tff(fact_10479_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qw(Uu),Uua) = suminf_real(aa_nat_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_278
tff(fact_10480_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_qv(Uu),Uua) = suminf_nat(aa_nat_fun_nat_nat(Uu,Uua)) ).

% ATP.lambda_279
tff(fact_10481_ATP_Olambda__280,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: int] : aa_int_complex(aTP_Lamm_qs(Uu),Uua) = suminf_complex(aa_int4144929030551567077omplex(Uu,Uua)) ).

% ATP.lambda_280
tff(fact_10482_ATP_Olambda__281,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: int] : aa_int_real(aTP_Lamm_qp(Uu),Uua) = suminf_real(aa_int_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_281
tff(fact_10483_ATP_Olambda__282,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: int] : aa_int_int(aTP_Lamm_qt(Uu),Uua) = suminf_int(aa_int_fun_nat_int(Uu,Uua)) ).

% ATP.lambda_282
tff(fact_10484_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_abb(Uu),Uua) = aa_complex_complex(cnj,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_283
tff(fact_10485_ATP_Olambda__284,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ky(Uu),Uua)
    <=> ~ aa_Pro4592068669816069595_int_o(Uu,Uua) ) ).

% ATP.lambda_284
tff(fact_10486_ATP_Olambda__285,axiom,
    ! [Uu: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kz(Uu),Uua)
    <=> ~ aa_complex_o(Uu,Uua) ) ).

% ATP.lambda_285
tff(fact_10487_ATP_Olambda__286,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_la(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_286
tff(fact_10488_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_lb(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_287
tff(fact_10489_ATP_Olambda__288,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_lc(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_288
tff(fact_10490_ATP_Olambda__289,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tp(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_289
tff(fact_10491_ATP_Olambda__290,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sh(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_290
tff(fact_10492_ATP_Olambda__291,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fa(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_291
tff(fact_10493_ATP_Olambda__292,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ey(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_292
tff(fact_10494_ATP_Olambda__293,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ez(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_293
tff(fact_10495_ATP_Olambda__294,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_oy(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_294
tff(fact_10496_ATP_Olambda__295,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oz(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_295
tff(fact_10497_ATP_Olambda__296,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_pa(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_296
tff(fact_10498_ATP_Olambda__297,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pb(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_297
tff(fact_10499_ATP_Olambda__298,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fh(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_298
tff(fact_10500_ATP_Olambda__299,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fl(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_299
tff(fact_10501_ATP_Olambda__300,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_fc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_300
tff(fact_10502_ATP_Olambda__301,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ff(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_301
tff(fact_10503_ATP_Olambda__302,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fi(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_302
tff(fact_10504_ATP_Olambda__303,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_303
tff(fact_10505_ATP_Olambda__304,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fj(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_304
tff(fact_10506_ATP_Olambda__305,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_fe(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_305
tff(fact_10507_ATP_Olambda__306,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fg(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_306
tff(fact_10508_ATP_Olambda__307,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fk(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_307
tff(fact_10509_ATP_Olambda__308,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fr(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_308
tff(fact_10510_ATP_Olambda__309,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fv(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_309
tff(fact_10511_ATP_Olambda__310,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_fm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_310
tff(fact_10512_ATP_Olambda__311,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_311
tff(fact_10513_ATP_Olambda__312,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fs(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_312
tff(fact_10514_ATP_Olambda__313,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_313
tff(fact_10515_ATP_Olambda__314,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_314
tff(fact_10516_ATP_Olambda__315,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ft(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_315
tff(fact_10517_ATP_Olambda__316,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_op(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_316
tff(fact_10518_ATP_Olambda__317,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_oq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_317
tff(fact_10519_ATP_Olambda__318,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_fo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_318
tff(fact_10520_ATP_Olambda__319,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_319
tff(fact_10521_ATP_Olambda__320,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fu(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_320
tff(fact_10522_ATP_Olambda__321,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_abm(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_321
tff(fact_10523_ATP_Olambda__322,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aci(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_322
tff(fact_10524_ATP_Olambda__323,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_abl(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_323
tff(fact_10525_ATP_Olambda__324,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ou(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_324
tff(fact_10526_ATP_Olambda__325,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ov(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_325
tff(fact_10527_ATP_Olambda__326,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ow(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_326
tff(fact_10528_ATP_Olambda__327,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ox(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_327
tff(fact_10529_ATP_Olambda__328,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_ju(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_328
tff(fact_10530_ATP_Olambda__329,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_jy(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_329
tff(fact_10531_ATP_Olambda__330,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_jp(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_330
tff(fact_10532_ATP_Olambda__331,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_js(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_331
tff(fact_10533_ATP_Olambda__332,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_jv(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_332
tff(fact_10534_ATP_Olambda__333,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_jq(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_333
tff(fact_10535_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_jw(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_334
tff(fact_10536_ATP_Olambda__335,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_jr(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu,Uub),zero_zero_complex) ).

% ATP.lambda_335
tff(fact_10537_ATP_Olambda__336,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_jt(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_336
tff(fact_10538_ATP_Olambda__337,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_jx(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_337
tff(fact_10539_ATP_Olambda__338,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_aez(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_338
tff(fact_10540_ATP_Olambda__339,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_uz(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_339
tff(fact_10541_ATP_Olambda__340,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ux(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_340
tff(fact_10542_ATP_Olambda__341,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_gd(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_341
tff(fact_10543_ATP_Olambda__342,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: nat,Uub: complex] : aa_complex_complex(aa_nat3829984526574172007omplex(aTP_Lamm_pz(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(Uu,Uub),Uua) ).

% ATP.lambda_342
tff(fact_10544_ATP_Olambda__343,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: nat,Uub: complex] : aa_complex_real(aa_nat480766339200631653x_real(aTP_Lamm_pm(Uu),Uua),Uub) = aa_nat_real(aa_com20334259897932645t_real(Uu,Uub),Uua) ).

% ATP.lambda_343
tff(fact_10545_ATP_Olambda__344,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_jd(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_344
tff(fact_10546_ATP_Olambda__345,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ie(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_345
tff(fact_10547_ATP_Olambda__346,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_ij(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_346
tff(fact_10548_ATP_Olambda__347,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: nat,Uub: real] : aa_real_complex(aa_nat7791439356240649701omplex(aTP_Lamm_pr(Uu),Uua),Uub) = aa_nat_complex(aa_rea1531280491672885477omplex(Uu,Uub),Uua) ).

% ATP.lambda_347
tff(fact_10549_ATP_Olambda__348,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_jg(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_348
tff(fact_10550_ATP_Olambda__349,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_iv(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_349
tff(fact_10551_ATP_Olambda__350,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_hw(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_350
tff(fact_10552_ATP_Olambda__351,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_in(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_351
tff(fact_10553_ATP_Olambda__352,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pt(Uu),Uua),Uub) = aa_nat_complex(aa_nat6454746962644518025omplex(Uu,Uub),Uua) ).

% ATP.lambda_352
tff(fact_10554_ATP_Olambda__353,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gj(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_353
tff(fact_10555_ATP_Olambda__354,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_354
tff(fact_10556_ATP_Olambda__355,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ia(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_355
tff(fact_10557_ATP_Olambda__356,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_ir(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_356
tff(fact_10558_ATP_Olambda__357,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: nat,Uub: int] : aa_int_complex(aa_nat7430282427850695397omplex(aTP_Lamm_pv(Uu),Uua),Uub) = aa_nat_complex(aa_int4144929030551567077omplex(Uu,Uub),Uua) ).

% ATP.lambda_357
tff(fact_10559_ATP_Olambda__358,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_pp(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_358
tff(fact_10560_ATP_Olambda__359,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_iz(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_359
tff(fact_10561_ATP_Olambda__360,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_px(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_360
tff(fact_10562_ATP_Olambda__361,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ga(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_361
tff(fact_10563_ATP_Olambda__362,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_zt(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_zs(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_362
tff(fact_10564_ATP_Olambda__363,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zr(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_zq(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_363
tff(fact_10565_ATP_Olambda__364,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_zp(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_zo(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_364
tff(fact_10566_ATP_Olambda__365,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zn(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_zm(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_365
tff(fact_10567_ATP_Olambda__366,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vl(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_vk(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_366
tff(fact_10568_ATP_Olambda__367,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vj(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_vi(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_367
tff(fact_10569_ATP_Olambda__368,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_te(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,
          $ite(Uub = Uu,one_one_complex,zero_zero_complex)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_368
tff(fact_10570_ATP_Olambda__369,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tf(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_369
tff(fact_10571_ATP_Olambda__370,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_tg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu,one_one_int,zero_zero_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uub)) ).

% ATP.lambda_370
tff(fact_10572_ATP_Olambda__371,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_abj(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_371
tff(fact_10573_ATP_Olambda__372,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_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gj(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_372
tff(fact_10574_ATP_Olambda__373,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_gg(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_373
tff(fact_10575_ATP_Olambda__374,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_acp((Uu)),Uua),(Uub)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_aco((Uu)),(Uub)),code_bit_cut_integer(Uua)) ).

% ATP.lambda_374
tff(fact_10576_ATP_Olambda__375,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aax(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aaw(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_375
tff(fact_10577_ATP_Olambda__376,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aav(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aau(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_376
tff(fact_10578_ATP_Olambda__377,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aat(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aas(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_377
tff(fact_10579_ATP_Olambda__378,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aar(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aaq(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_378
tff(fact_10580_ATP_Olambda__379,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aap(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aao(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_379
tff(fact_10581_ATP_Olambda__380,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aan(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aam(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_380
tff(fact_10582_ATP_Olambda__381,axiom,
    ! [Uu: set_complex,Uua: fun_co3258106519541737638omplex,Uub: nat] : aa_nat_complex(aa_fun5003373910569599327omplex(aTP_Lamm_qa(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(aTP_Lamm_pz(Uua),Uub),Uu) ).

% ATP.lambda_381
tff(fact_10583_ATP_Olambda__382,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: nat] : aa_nat_int(aa_fun8710705519803144921at_int(aTP_Lamm_py(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_nat_fun_int_int(aTP_Lamm_px(Uua),Uub),Uu) ).

% ATP.lambda_382
tff(fact_10584_ATP_Olambda__383,axiom,
    ! [Uu: set_int,Uua: fun_in8056368165505483812omplex,Uub: nat] : aa_nat_complex(aa_fun3543561798013398749omplex(aTP_Lamm_pw(Uu),Uua),Uub) = groups3049146728041665814omplex(aa_nat7430282427850695397omplex(aTP_Lamm_pv(Uua),Uub),Uu) ).

% ATP.lambda_383
tff(fact_10585_ATP_Olambda__384,axiom,
    ! [Uu: set_nat,Uua: fun_na1142814060743658952omplex,Uub: nat] : aa_nat_complex(aa_fun4886358607025424513omplex(aTP_Lamm_pu(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_pt(Uua),Uub)),Uu) ).

% ATP.lambda_384
tff(fact_10586_ATP_Olambda__385,axiom,
    ! [Uu: set_real,Uua: fun_re2084383175013312036omplex,Uub: nat] : aa_nat_complex(aa_fun7632768522510439005omplex(aTP_Lamm_ps(Uu),Uua),Uub) = groups5754745047067104278omplex(aa_nat7791439356240649701omplex(aTP_Lamm_pr(Uua),Uub),Uu) ).

% ATP.lambda_385
tff(fact_10587_ATP_Olambda__386,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun2842512154077127769t_real(aTP_Lamm_pq(Uu),Uua),Uub) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_pp(Uua),Uub),Uu) ).

% ATP.lambda_386
tff(fact_10588_ATP_Olambda__387,axiom,
    ! [Uu: set_complex,Uua: fun_co8378349014452122532t_real,Uub: nat] : aa_nat_real(aa_fun471735231769099483t_real(aTP_Lamm_pn(Uu),Uua),Uub) = groups5808333547571424918x_real(aa_nat480766339200631653x_real(aTP_Lamm_pm(Uua),Uub),Uu) ).

% ATP.lambda_387
tff(fact_10589_ATP_Olambda__388,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: nat] : aa_nat_real(aa_fun628238740774635481t_real(aTP_Lamm_po(Uu),Uua),Uub) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_jg(Uua),Uub),Uu) ).

% ATP.lambda_388
tff(fact_10590_ATP_Olambda__389,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_gk(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gj(Uu),Uub)),Uua) ).

% ATP.lambda_389
tff(fact_10591_ATP_Olambda__390,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_qc(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gj(Uua),Uub)),Uu) ).

% ATP.lambda_390
tff(fact_10592_ATP_Olambda__391,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gh(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gg(Uu),Uub)),Uua) ).

% ATP.lambda_391
tff(fact_10593_ATP_Olambda__392,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_qb(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gg(Uua),Uub)),Uu) ).

% ATP.lambda_392
tff(fact_10594_ATP_Olambda__393,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_ge(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_gd(Uu),Uub),Uua) ).

% ATP.lambda_393
tff(fact_10595_ATP_Olambda__394,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_gb(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ga(Uu),Uub),Uua) ).

% ATP.lambda_394
tff(fact_10596_ATP_Olambda__395,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ada(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_395
tff(fact_10597_ATP_Olambda__396,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_sr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_396
tff(fact_10598_ATP_Olambda__397,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ss(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_397
tff(fact_10599_ATP_Olambda__398,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_st(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_398
tff(fact_10600_ATP_Olambda__399,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_su(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_399
tff(fact_10601_ATP_Olambda__400,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_sv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_400
tff(fact_10602_ATP_Olambda__401,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_vd(Uu),Uua),Uub)
    <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ql(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_401
tff(fact_10603_ATP_Olambda__402,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_ve(Uu),Uua),Uub)
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qj(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_402
tff(fact_10604_ATP_Olambda__403,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ks(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_403
tff(fact_10605_ATP_Olambda__404,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ku(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_404
tff(fact_10606_ATP_Olambda__405,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kt(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_405
tff(fact_10607_ATP_Olambda__406,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_406
tff(fact_10608_ATP_Olambda__407,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_sw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_407
tff(fact_10609_ATP_Olambda__408,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tu(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(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_408
tff(fact_10610_ATP_Olambda__409,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(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_409
tff(fact_10611_ATP_Olambda__410,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_bw(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_410
tff(fact_10612_ATP_Olambda__411,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( ord_le211207098394363844omplex(set_complex2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_411
tff(fact_10613_ATP_Olambda__412,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_412
tff(fact_10614_ATP_Olambda__413,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_413
tff(fact_10615_ATP_Olambda__414,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_by(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_int(set_int2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_414
tff(fact_10616_ATP_Olambda__415,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_br(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_415
tff(fact_10617_ATP_Olambda__416,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( ord_le211207098394363844omplex(set_complex2(Uub),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_416
tff(fact_10618_ATP_Olambda__417,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_417
tff(fact_10619_ATP_Olambda__418,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_418
tff(fact_10620_ATP_Olambda__419,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_int(set_int2(Uub),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_419
tff(fact_10621_ATP_Olambda__420,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_abd(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_420
tff(fact_10622_ATP_Olambda__421,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_abn(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_421
tff(fact_10623_ATP_Olambda__422,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_abe(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_422
tff(fact_10624_ATP_Olambda__423,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_tt(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_423
tff(fact_10625_ATP_Olambda__424,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_424
tff(fact_10626_ATP_Olambda__425,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ub(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uub)) ).

% ATP.lambda_425
tff(fact_10627_ATP_Olambda__426,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ua(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_426
tff(fact_10628_ATP_Olambda__427,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vb(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_427
tff(fact_10629_ATP_Olambda__428,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_zy(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_428
tff(fact_10630_ATP_Olambda__429,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_429
tff(fact_10631_ATP_Olambda__430,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_zz(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_430
tff(fact_10632_ATP_Olambda__431,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ct(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_431
tff(fact_10633_ATP_Olambda__432,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cs(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_432
tff(fact_10634_ATP_Olambda__433,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cr(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_433
tff(fact_10635_ATP_Olambda__434,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ak(Uu),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_434
tff(fact_10636_ATP_Olambda__435,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_nat_o(vEBT_vebt_member(Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_435
tff(fact_10637_ATP_Olambda__436,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aeo(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_436
tff(fact_10638_ATP_Olambda__437,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aeu(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uua)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uu) ) ) ).

% ATP.lambda_437
tff(fact_10639_ATP_Olambda__438,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_bd(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_438
tff(fact_10640_ATP_Olambda__439,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_439
tff(fact_10641_ATP_Olambda__440,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bc(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_440
tff(fact_10642_ATP_Olambda__441,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_441
tff(fact_10643_ATP_Olambda__442,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_442
tff(fact_10644_ATP_Olambda__443,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_443
tff(fact_10645_ATP_Olambda__444,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_ef(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_444
tff(fact_10646_ATP_Olambda__445,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_en(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_445
tff(fact_10647_ATP_Olambda__446,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_dz(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_446
tff(fact_10648_ATP_Olambda__447,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_eh(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_447
tff(fact_10649_ATP_Olambda__448,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_ep(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_448
tff(fact_10650_ATP_Olambda__449,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_eb(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_449
tff(fact_10651_ATP_Olambda__450,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_ej(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_450
tff(fact_10652_ATP_Olambda__451,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_er(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_451
tff(fact_10653_ATP_Olambda__452,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_ed(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).

% ATP.lambda_452
tff(fact_10654_ATP_Olambda__453,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_el(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_453
tff(fact_10655_ATP_Olambda__454,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_dl(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_454
tff(fact_10656_ATP_Olambda__455,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_dt(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_455
tff(fact_10657_ATP_Olambda__456,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_df(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_456
tff(fact_10658_ATP_Olambda__457,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_dn(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_457
tff(fact_10659_ATP_Olambda__458,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_dv(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_458
tff(fact_10660_ATP_Olambda__459,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_dh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_459
tff(fact_10661_ATP_Olambda__460,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_dp(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_460
tff(fact_10662_ATP_Olambda__461,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_dx(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_461
tff(fact_10663_ATP_Olambda__462,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_dj(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_462
tff(fact_10664_ATP_Olambda__463,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_dr(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_463
tff(fact_10665_ATP_Olambda__464,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_464
tff(fact_10666_ATP_Olambda__465,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & ~ member_complex(Uub,Uua) ) ) ).

% ATP.lambda_465
tff(fact_10667_ATP_Olambda__466,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ~ member_real(Uub,Uua) ) ) ).

% ATP.lambda_466
tff(fact_10668_ATP_Olambda__467,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & ~ member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_467
tff(fact_10669_ATP_Olambda__468,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_av(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_468
tff(fact_10670_ATP_Olambda__469,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ~ member_int(Uub,Uua) ) ) ).

% ATP.lambda_469
tff(fact_10671_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aen(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_470
tff(fact_10672_ATP_Olambda__471,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_uw(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_471
tff(fact_10673_ATP_Olambda__472,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_afk(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_472
tff(fact_10674_ATP_Olambda__473,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_afi(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu) ) ).

% ATP.lambda_473
tff(fact_10675_ATP_Olambda__474,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vh(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_474
tff(fact_10676_ATP_Olambda__475,axiom,
    ! [Uu: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_oj(Uu),Uua),(Uub))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(Uua,(Uub)),Uu) ) ).

% ATP.lambda_475
tff(fact_10677_ATP_Olambda__476,axiom,
    ! [Uu: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_ok(Uu),Uua),Uub)
    <=> member7279096912039735102um_num(product_Pair_num_num(Uua,Uub),Uu) ) ).

% ATP.lambda_476
tff(fact_10678_ATP_Olambda__477,axiom,
    ! [Uu: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_ol(Uu),Uua),Uub)
    <=> member9148766508732265716at_num(product_Pair_nat_num(Uua,Uub),Uu) ) ).

% ATP.lambda_477
tff(fact_10679_ATP_Olambda__478,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_om(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_478
tff(fact_10680_ATP_Olambda__479,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_on(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_479
tff(fact_10681_ATP_Olambda__480,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_de(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_480
tff(fact_10682_ATP_Olambda__481,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_aaa(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_481
tff(fact_10683_ATP_Olambda__482,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qy(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_482
tff(fact_10684_ATP_Olambda__483,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_483
tff(fact_10685_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_aeg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_484
tff(fact_10686_ATP_Olambda__485,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_acy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_485
tff(fact_10687_ATP_Olambda__486,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ql(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_486
tff(fact_10688_ATP_Olambda__487,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_487
tff(fact_10689_ATP_Olambda__488,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_vf(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_488
tff(fact_10690_ATP_Olambda__489,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vy(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_489
tff(fact_10691_ATP_Olambda__490,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_vg(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uub)) ).

% ATP.lambda_490
tff(fact_10692_ATP_Olambda__491,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_491
tff(fact_10693_ATP_Olambda__492,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afa(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_492
tff(fact_10694_ATP_Olambda__493,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_afc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_493
tff(fact_10695_ATP_Olambda__494,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aay(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_494
tff(fact_10696_ATP_Olambda__495,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_me(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_495
tff(fact_10697_ATP_Olambda__496,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_mi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_496
tff(fact_10698_ATP_Olambda__497,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ml(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_497
tff(fact_10699_ATP_Olambda__498,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_mf(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_498
tff(fact_10700_ATP_Olambda__499,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_499
tff(fact_10701_ATP_Olambda__500,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_mm(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_500
tff(fact_10702_ATP_Olambda__501,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_mg(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_501
tff(fact_10703_ATP_Olambda__502,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mn(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_502
tff(fact_10704_ATP_Olambda__503,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_mh(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_503
tff(fact_10705_ATP_Olambda__504,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_mk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_504
tff(fact_10706_ATP_Olambda__505,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_hq(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_505
tff(fact_10707_ATP_Olambda__506,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_pj(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_506
tff(fact_10708_ATP_Olambda__507,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_507
tff(fact_10709_ATP_Olambda__508,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hp(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_508
tff(fact_10710_ATP_Olambda__509,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_lj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).

% ATP.lambda_509
tff(fact_10711_ATP_Olambda__510,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_lk(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_510
tff(fact_10712_ATP_Olambda__511,axiom,
    ! [Uu: fun_o_nat,Uua: fun_o_nat,Uub: $o] : aa_o_nat(aa_fun6176012476734143100_o_nat(aTP_Lamm_re(Uu),Uua),(Uub)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_o_nat(Uua,(Uub))),aa_o_nat(Uu,(Uub))) ).

% ATP.lambda_511
tff(fact_10713_ATP_Olambda__512,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_ll(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_512
tff(fact_10714_ATP_Olambda__513,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_th(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(Uua,Uub)),aa_nat_complex(Uu,Uub)) ).

% ATP.lambda_513
tff(fact_10715_ATP_Olambda__514,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_514
tff(fact_10716_ATP_Olambda__515,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ln(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_515
tff(fact_10717_ATP_Olambda__516,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_lm(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_516
tff(fact_10718_ATP_Olambda__517,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_hm(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_517
tff(fact_10719_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ph(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_518
tff(fact_10720_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ho(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_519
tff(fact_10721_ATP_Olambda__520,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hn(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_520
tff(fact_10722_ATP_Olambda__521,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_pi(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_521
tff(fact_10723_ATP_Olambda__522,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_hl(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_522
tff(fact_10724_ATP_Olambda__523,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_acw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_523
tff(fact_10725_ATP_Olambda__524,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        | aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_524
tff(fact_10726_ATP_Olambda__525,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_525
tff(fact_10727_ATP_Olambda__526,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_526
tff(fact_10728_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_527
tff(fact_10729_ATP_Olambda__528,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cj(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_528
tff(fact_10730_ATP_Olambda__529,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_529
tff(fact_10731_ATP_Olambda__530,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_530
tff(fact_10732_ATP_Olambda__531,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_531
tff(fact_10733_ATP_Olambda__532,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_532
tff(fact_10734_ATP_Olambda__533,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_533
tff(fact_10735_ATP_Olambda__534,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_gc(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_534
tff(fact_10736_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_gi(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_535
tff(fact_10737_ATP_Olambda__536,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_gf(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_536
tff(fact_10738_ATP_Olambda__537,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_fz(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_537
tff(fact_10739_ATP_Olambda__538,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_538
tff(fact_10740_ATP_Olambda__539,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_539
tff(fact_10741_ATP_Olambda__540,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ji(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_540
tff(fact_10742_ATP_Olambda__541,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_hs(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_541
tff(fact_10743_ATP_Olambda__542,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ot(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_542
tff(fact_10744_ATP_Olambda__543,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ht(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_543
tff(fact_10745_ATP_Olambda__544,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_he(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_544
tff(fact_10746_ATP_Olambda__545,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pe(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_545
tff(fact_10747_ATP_Olambda__546,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_546
tff(fact_10748_ATP_Olambda__547,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hf(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_547
tff(fact_10749_ATP_Olambda__548,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_hd(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_548
tff(fact_10750_ATP_Olambda__549,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ta(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_549
tff(fact_10751_ATP_Olambda__550,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_550
tff(fact_10752_ATP_Olambda__551,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_rk(Uu),Uua),Uub) = minus_8373710615458151222nteger(aa_nat_Code_integer(Uu,Uub),Uua) ).

% ATP.lambda_551
tff(fact_10753_ATP_Olambda__552,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_552
tff(fact_10754_ATP_Olambda__553,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_ri(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uub)),Uua) ).

% ATP.lambda_553
tff(fact_10755_ATP_Olambda__554,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_rj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_554
tff(fact_10756_ATP_Olambda__555,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_act(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_555
tff(fact_10757_ATP_Olambda__556,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ael(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_556
tff(fact_10758_ATP_Olambda__557,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_acv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_557
tff(fact_10759_ATP_Olambda__558,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_rm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_558
tff(fact_10760_ATP_Olambda__559,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_nl(Uu),Uua),Uub) = plus_p5714425477246183910nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_559
tff(fact_10761_ATP_Olambda__560,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ni(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_560
tff(fact_10762_ATP_Olambda__561,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nn(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_561
tff(fact_10763_ATP_Olambda__562,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_nj(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_562
tff(fact_10764_ATP_Olambda__563,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nm(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_563
tff(fact_10765_ATP_Olambda__564,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_nk(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_564
tff(fact_10766_ATP_Olambda__565,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_sd(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_565
tff(fact_10767_ATP_Olambda__566,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sg(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_566
tff(fact_10768_ATP_Olambda__567,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_se(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_567
tff(fact_10769_ATP_Olambda__568,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sf(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_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_568
tff(fact_10770_ATP_Olambda__569,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aea(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_569
tff(fact_10771_ATP_Olambda__570,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ls(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_570
tff(fact_10772_ATP_Olambda__571,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lv(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_571
tff(fact_10773_ATP_Olambda__572,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_lt(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_572
tff(fact_10774_ATP_Olambda__573,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lu(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_573
tff(fact_10775_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mz(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_574
tff(fact_10776_ATP_Olambda__575,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aey(Uu),Uua),Uub)
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_575
tff(fact_10777_ATP_Olambda__576,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_hi(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_576
tff(fact_10778_ATP_Olambda__577,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_os(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_577
tff(fact_10779_ATP_Olambda__578,axiom,
    ! [Uu: 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(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_578
tff(fact_10780_ATP_Olambda__579,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_579
tff(fact_10781_ATP_Olambda__580,axiom,
    ! [Uu: 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(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_580
tff(fact_10782_ATP_Olambda__581,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pf(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uua),aa_nat_complex(Uu,Uub)) ).

% ATP.lambda_581
tff(fact_10783_ATP_Olambda__582,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_582
tff(fact_10784_ATP_Olambda__583,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_rd(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_583
tff(fact_10785_ATP_Olambda__584,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc(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_584
tff(fact_10786_ATP_Olambda__585,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aev(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_585
tff(fact_10787_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_or(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_586
tff(fact_10788_ATP_Olambda__587,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lr(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_587
tff(fact_10789_ATP_Olambda__588,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aet(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_588
tff(fact_10790_ATP_Olambda__589,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lq(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_589
tff(fact_10791_ATP_Olambda__590,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_jk(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_590
tff(fact_10792_ATP_Olambda__591,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_jl(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_591
tff(fact_10793_ATP_Olambda__592,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jm(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_592
tff(fact_10794_ATP_Olambda__593,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jn(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_593
tff(fact_10795_ATP_Olambda__594,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jo(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_594
tff(fact_10796_ATP_Olambda__595,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_tc(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_595
tff(fact_10797_ATP_Olambda__596,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_td(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_596
tff(fact_10798_ATP_Olambda__597,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_qm(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub))) ).

% ATP.lambda_597
tff(fact_10799_ATP_Olambda__598,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qk(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_598
tff(fact_10800_ATP_Olambda__599,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aef(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_599
tff(fact_10801_ATP_Olambda__600,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_zo(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_600
tff(fact_10802_ATP_Olambda__601,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_zm(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_601
tff(fact_10803_ATP_Olambda__602,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_zs(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_rea2618301233319580611omplex(real_V2046097035970521341omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_complex) ).

% ATP.lambda_602
tff(fact_10804_ATP_Olambda__603,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_zq(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_real) ).

% ATP.lambda_603
tff(fact_10805_ATP_Olambda__604,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wc(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_604
tff(fact_10806_ATP_Olambda__605,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_wk(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_605
tff(fact_10807_ATP_Olambda__606,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_we(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(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_606
tff(fact_10808_ATP_Olambda__607,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wi(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_607
tff(fact_10809_ATP_Olambda__608,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wg(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_608
tff(fact_10810_ATP_Olambda__609,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_wd(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_609
tff(fact_10811_ATP_Olambda__610,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) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_610
tff(fact_10812_ATP_Olambda__611,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_wf(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_611
tff(fact_10813_ATP_Olambda__612,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_wj(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_612
tff(fact_10814_ATP_Olambda__613,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_wh(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_613
tff(fact_10815_ATP_Olambda__614,axiom,
    ! [Uu: fun_nat_complex,Uua: set_nat,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_set5115289136882354743omplex(aTP_Lamm_ti(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_complex(Uub,Uuc),aa_nat_complex(Uu,Uuc)) ).

% ATP.lambda_614
tff(fact_10816_ATP_Olambda__615,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_tk(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_615
tff(fact_10817_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_oi(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_oh(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_616
tff(fact_10818_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_og(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_of(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_617
tff(fact_10819_ATP_Olambda__618,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_od(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_618
tff(fact_10820_ATP_Olambda__619,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_aco((Uu)),(Uua)),Uub),(Uuc)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_acn((Uu)),(Uua)),(Uuc)),code_bit_cut_integer(Uub)) ).

% ATP.lambda_619
tff(fact_10821_ATP_Olambda__620,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_hc(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_hb(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_620
tff(fact_10822_ATP_Olambda__621,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_ha(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_gz(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_621
tff(fact_10823_ATP_Olambda__622,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_gy(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_gx(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_622
tff(fact_10824_ATP_Olambda__623,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_gw(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_gv(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_623
tff(fact_10825_ATP_Olambda__624,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_wp(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          $ite(
            Uuc = zero_zero_nat,
            aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),
            $ite(Uuc = Uu,one_one_Code_integer,zero_z3403309356797280102nteger) )),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_624
tff(fact_10826_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_wn(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,
          $ite(
            Uuc = zero_zero_nat,
            aa_complex_complex(uminus1482373934393186551omplex,Uub),
            $ite(Uuc = Uu,one_one_complex,zero_zero_complex) )),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uuc)) ).

% ATP.lambda_625
tff(fact_10827_ATP_Olambda__626,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] :
      aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_wq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(
            Uuc = zero_zero_nat,
            aa_real_real(uminus_uminus_real,Uub),
            $ite(Uuc = Uu,one_one_real,zero_zero_real) )),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_626
tff(fact_10828_ATP_Olambda__627,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] :
      aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_wo(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat($ite(
            Uuc = zero_zero_nat,
            aa_rat_rat(uminus_uminus_rat,Uub),
            $ite(Uuc = Uu,one_one_rat,zero_zero_rat) )),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_627
tff(fact_10829_ATP_Olambda__628,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_wm(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(
            Uuc = zero_zero_nat,
            aa_int_int(uminus_uminus_int,Uub),
            $ite(Uuc = Uu,one_one_int,zero_zero_int) )),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_628
tff(fact_10830_ATP_Olambda__629,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_jh(Uu),Uua),Uub),Uuc) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_jg(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_cu(Uu),Uub),Uuc))) ).

% ATP.lambda_629
tff(fact_10831_ATP_Olambda__630,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_je(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_jd(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_dd(Uu),Uub),Uuc))) ).

% ATP.lambda_630
tff(fact_10832_ATP_Olambda__631,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ja(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_iz(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_da(Uu),Uub),Uuc))) ).

% ATP.lambda_631
tff(fact_10833_ATP_Olambda__632,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_iw(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_iv(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_cu(Uu),Uub),Uuc))) ).

% ATP.lambda_632
tff(fact_10834_ATP_Olambda__633,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_is(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_ir(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_dc(Uu),Uub),Uuc))) ).

% ATP.lambda_633
tff(fact_10835_ATP_Olambda__634,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_io(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_in(Uua),Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_cz(Uu),Uub),Uuc))) ).

% ATP.lambda_634
tff(fact_10836_ATP_Olambda__635,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_ik(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_ij(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_cw(Uu),Uub),Uuc))) ).

% ATP.lambda_635
tff(fact_10837_ATP_Olambda__636,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: int] : aa_int_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_ig(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_ie(Uua),Uuc),collect_complex(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_if(Uu),Uub),Uuc))) ).

% ATP.lambda_636
tff(fact_10838_ATP_Olambda__637,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_ib(Uu),Uua),Uub),Uuc) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ia(Uua),Uuc),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_cy(Uu),Uub),Uuc))) ).

% ATP.lambda_637
tff(fact_10839_ATP_Olambda__638,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: int] : aa_int_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_hx(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_hw(Uua),Uuc),collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_cv(Uu),Uub),Uuc))) ).

% ATP.lambda_638
tff(fact_10840_ATP_Olambda__639,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_wa(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_vz(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_639
tff(fact_10841_ATP_Olambda__640,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_vr(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_vq(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),Uuc)) ).

% ATP.lambda_640
tff(fact_10842_ATP_Olambda__641,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_vp(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_vo(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_641
tff(fact_10843_ATP_Olambda__642,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_vs(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_vk(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_642
tff(fact_10844_ATP_Olambda__643,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_vn(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_vi(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uuc)) ).

% ATP.lambda_643
tff(fact_10845_ATP_Olambda__644,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_adf(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_644
tff(fact_10846_ATP_Olambda__645,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_add(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_645
tff(fact_10847_ATP_Olambda__646,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_adb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_646
tff(fact_10848_ATP_Olambda__647,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_adc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_647
tff(fact_10849_ATP_Olambda__648,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_rn(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uub)) ).

% ATP.lambda_648
tff(fact_10850_ATP_Olambda__649,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_rt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_649
tff(fact_10851_ATP_Olambda__650,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rp(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_650
tff(fact_10852_ATP_Olambda__651,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rr(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(aa_int_fun_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(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uub)) ).

% ATP.lambda_651
tff(fact_10853_ATP_Olambda__652,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abp(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_652
tff(fact_10854_ATP_Olambda__653,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_od(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uua),Uu)),Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(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_653
tff(fact_10855_ATP_Olambda__654,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_654
tff(fact_10856_ATP_Olambda__655,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_of(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_655
tff(fact_10857_ATP_Olambda__656,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_yj(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Uu),Uuc))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_656
tff(fact_10858_ATP_Olambda__657,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_yk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_657
tff(fact_10859_ATP_Olambda__658,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_yh(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_658
tff(fact_10860_ATP_Olambda__659,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_yi(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(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_659
tff(fact_10861_ATP_Olambda__660,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat,Uuc: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aa_Cod3140087502949435154nteger(aTP_Lamm_vw(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uu),Uuc))),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_660
tff(fact_10862_ATP_Olambda__661,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_vt(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_661
tff(fact_10863_ATP_Olambda__662,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_vx(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_662
tff(fact_10864_ATP_Olambda__663,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_vu(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_663
tff(fact_10865_ATP_Olambda__664,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vm(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_664
tff(fact_10866_ATP_Olambda__665,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_vv(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(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_665
tff(fact_10867_ATP_Olambda__666,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_rz(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)) ).

% ATP.lambda_666
tff(fact_10868_ATP_Olambda__667,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_667
tff(fact_10869_ATP_Olambda__668,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_sa(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_668
tff(fact_10870_ATP_Olambda__669,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_sb(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_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(aa_int_fun_nat_int(power_power_int,Uu),Uuc)) ).

% ATP.lambda_669
tff(fact_10871_ATP_Olambda__670,axiom,
    ! [Uu: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_ih(Uu),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_670
tff(fact_10872_ATP_Olambda__671,axiom,
    ! [Uu: set_complex,Uua: fun_na1554462459066888140plex_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_il(Uu),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_671
tff(fact_10873_ATP_Olambda__672,axiom,
    ! [Uu: set_complex,Uua: fun_in5065666251602127472plex_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_ip(Uu),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_672
tff(fact_10874_ATP_Olambda__673,axiom,
    ! [Uu: set_nat,Uua: fun_co5676208753217392844_nat_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_jb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_673
tff(fact_10875_ATP_Olambda__674,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_it(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_674
tff(fact_10876_ATP_Olambda__675,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_ix(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_675
tff(fact_10877_ATP_Olambda__676,axiom,
    ! [Uu: set_int,Uua: fun_co3431063785546946928_int_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_ic(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_676
tff(fact_10878_ATP_Olambda__677,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_o,Uub: real,Uuc: int] :
      ( aa_int_o(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_hu(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_677
tff(fact_10879_ATP_Olambda__678,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_hy(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_678
tff(fact_10880_ATP_Olambda__679,axiom,
    ! [Uu: set_complex,Uua: fun_co5676208753217392844_nat_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_dd(Uu),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_679
tff(fact_10881_ATP_Olambda__680,axiom,
    ! [Uu: set_complex,Uua: fun_co3431063785546946928_int_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_if(Uu),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_680
tff(fact_10882_ATP_Olambda__681,axiom,
    ! [Uu: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_cw(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_681
tff(fact_10883_ATP_Olambda__682,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_cu(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_682
tff(fact_10884_ATP_Olambda__683,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_o,Uub: int,Uuc: real] :
      ( aa_real_o(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_cv(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_683
tff(fact_10885_ATP_Olambda__684,axiom,
    ! [Uu: set_nat,Uua: fun_na1554462459066888140plex_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_cz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_684
tff(fact_10886_ATP_Olambda__685,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_cx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_685
tff(fact_10887_ATP_Olambda__686,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_cy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_686
tff(fact_10888_ATP_Olambda__687,axiom,
    ! [Uu: set_int,Uua: fun_in5065666251602127472plex_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_dc(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_687
tff(fact_10889_ATP_Olambda__688,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_da(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_688
tff(fact_10890_ATP_Olambda__689,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_db(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_689
tff(fact_10891_ATP_Olambda__690,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ro(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_690
tff(fact_10892_ATP_Olambda__691,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ru(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_691
tff(fact_10893_ATP_Olambda__692,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_rq(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_692
tff(fact_10894_ATP_Olambda__693,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_rs(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_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(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_693
tff(fact_10895_ATP_Olambda__694,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_rv(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_694
tff(fact_10896_ATP_Olambda__695,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ry(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_695
tff(fact_10897_ATP_Olambda__696,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_rw(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_696
tff(fact_10898_ATP_Olambda__697,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_697
tff(fact_10899_ATP_Olambda__698,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_vc(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_698
tff(fact_10900_ATP_Olambda__699,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aao(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_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_699
tff(fact_10901_ATP_Olambda__700,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abu(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_700
tff(fact_10902_ATP_Olambda__701,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aam(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_701
tff(fact_10903_ATP_Olambda__702,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_abs(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_702
tff(fact_10904_ATP_Olambda__703,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_abw(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_703
tff(fact_10905_ATP_Olambda__704,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_aby(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_704
tff(fact_10906_ATP_Olambda__705,axiom,
    ! [Uu: set_complex,Uua: complex,Uub: complex,Uuc: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aa_com7221863818706917250plex_o(aTP_Lamm_afm(Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => member_complex(Uuc,Uu) ) ) ).

% ATP.lambda_705
tff(fact_10907_ATP_Olambda__706,axiom,
    ! [Uu: set_real,Uua: real,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_rea1652550732481586312real_o(aTP_Lamm_afn(Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => member_real(Uuc,Uu) ) ) ).

% ATP.lambda_706
tff(fact_10908_ATP_Olambda__707,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dm(Uu),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_707
tff(fact_10909_ATP_Olambda__708,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_du(Uu),Uua),Uub),Uuc)
    <=> ( 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_708
tff(fact_10910_ATP_Olambda__709,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_dg(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_709
tff(fact_10911_ATP_Olambda__710,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_do(Uu),Uua),Uub),Uuc)
    <=> ( 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_710
tff(fact_10912_ATP_Olambda__711,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_dw(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_711
tff(fact_10913_ATP_Olambda__712,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_di(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_712
tff(fact_10914_ATP_Olambda__713,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_713
tff(fact_10915_ATP_Olambda__714,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_714
tff(fact_10916_ATP_Olambda__715,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_dk(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_715
tff(fact_10917_ATP_Olambda__716,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_ds(Uu),Uua),Uub),Uuc)
    <=> ( 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_716
tff(fact_10918_ATP_Olambda__717,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_eg(Uu),Uua),Uub),Uuc)
    <=> ( member_complex(Uuc,Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_717
tff(fact_10919_ATP_Olambda__718,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_eo(Uu),Uua),Uub),Uuc)
    <=> ( 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_718
tff(fact_10920_ATP_Olambda__719,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_ea(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_719
tff(fact_10921_ATP_Olambda__720,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_ei(Uu),Uua),Uub),Uuc)
    <=> ( 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_720
tff(fact_10922_ATP_Olambda__721,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_eq(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_721
tff(fact_10923_ATP_Olambda__722,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_ec(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_722
tff(fact_10924_ATP_Olambda__723,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_ek(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_723
tff(fact_10925_ATP_Olambda__724,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_es(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_724
tff(fact_10926_ATP_Olambda__725,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_ee(Uu),Uua),Uub),Uuc)
    <=> ( member_int(Uuc,Uu)
        & ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).

% ATP.lambda_725
tff(fact_10927_ATP_Olambda__726,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_em(Uu),Uua),Uub),Uuc)
    <=> ( 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_726
tff(fact_10928_ATP_Olambda__727,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ra(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uuc)) ).

% ATP.lambda_727
tff(fact_10929_ATP_Olambda__728,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_728
tff(fact_10930_ATP_Olambda__729,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xk(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uuc)),aa_nat_complex(aa_com8181001469867462503omplex(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_729
tff(fact_10931_ATP_Olambda__730,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_xq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_730
tff(fact_10932_ATP_Olambda__731,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_xm(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_731
tff(fact_10933_ATP_Olambda__732,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_xo(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(aa_int_fun_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_732
tff(fact_10934_ATP_Olambda__733,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_vi(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(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_733
tff(fact_10935_ATP_Olambda__734,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_vk(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_734
tff(fact_10936_ATP_Olambda__735,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_vo(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_735
tff(fact_10937_ATP_Olambda__736,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_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_736
tff(fact_10938_ATP_Olambda__737,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_vq(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_737
tff(fact_10939_ATP_Olambda__738,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_gx(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_738
tff(fact_10940_ATP_Olambda__739,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_hb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_739
tff(fact_10941_ATP_Olambda__740,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_gz(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_740
tff(fact_10942_ATP_Olambda__741,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_gv(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_741
tff(fact_10943_ATP_Olambda__742,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_jc(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_jb(Uu),Uub),Uuc))) ).

% ATP.lambda_742
tff(fact_10944_ATP_Olambda__743,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_iy(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_ix(Uu),Uub),Uuc))) ).

% ATP.lambda_743
tff(fact_10945_ATP_Olambda__744,axiom,
    ! [Uu: set_nat,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_jf(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_it(Uu),Uub),Uuc))) ).

% ATP.lambda_744
tff(fact_10946_ATP_Olambda__745,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_iu(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_it(Uu),Uub),Uuc))) ).

% ATP.lambda_745
tff(fact_10947_ATP_Olambda__746,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_iq(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_ip(Uu),Uub),Uuc))) ).

% ATP.lambda_746
tff(fact_10948_ATP_Olambda__747,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_im(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_il(Uu),Uub),Uuc))) ).

% ATP.lambda_747
tff(fact_10949_ATP_Olambda__748,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_ii(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_ih(Uu),Uub),Uuc))) ).

% ATP.lambda_748
tff(fact_10950_ATP_Olambda__749,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: complex] : aa_complex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_id(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_ic(Uu),Uub),Uuc))) ).

% ATP.lambda_749
tff(fact_10951_ATP_Olambda__750,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_hz(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_hy(Uu),Uub),Uuc))) ).

% ATP.lambda_750
tff(fact_10952_ATP_Olambda__751,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: real] : aa_real_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_hv(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_hu(Uu),Uub),Uuc))) ).

% ATP.lambda_751
tff(fact_10953_ATP_Olambda__752,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_na((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_752
tff(fact_10954_ATP_Olambda__753,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_gt(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_753
tff(fact_10955_ATP_Olambda__754,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_gp(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_754
tff(fact_10956_ATP_Olambda__755,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7461304428735181397nt_int(aTP_Lamm_gq(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_755
tff(fact_10957_ATP_Olambda__756,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_gr(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_756
tff(fact_10958_ATP_Olambda__757,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_gl(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_757
tff(fact_10959_ATP_Olambda__758,axiom,
    ! [Uu: fun_o_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7323606162024495668nt_int(aTP_Lamm_gm(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_758
tff(fact_10960_ATP_Olambda__759,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_gs(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_759
tff(fact_10961_ATP_Olambda__760,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_gn(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_760
tff(fact_10962_ATP_Olambda__761,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun6119123525526402106nt_int(aTP_Lamm_go(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_761
tff(fact_10963_ATP_Olambda__762,axiom,
    ! [Uu: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_gu(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_762
tff(fact_10964_ATP_Olambda__763,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_lx(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_763
tff(fact_10965_ATP_Olambda__764,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_lw(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_764
tff(fact_10966_ATP_Olambda__765,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aaq(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_765
tff(fact_10967_ATP_Olambda__766,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aas(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_766
tff(fact_10968_ATP_Olambda__767,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aau(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_767
tff(fact_10969_ATP_Olambda__768,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aaw(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_768
tff(fact_10970_ATP_Olambda__769,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_xd(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_xc(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_769
tff(fact_10971_ATP_Olambda__770,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_xb(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(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_770
tff(fact_10972_ATP_Olambda__771,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_wz(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(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_771
tff(fact_10973_ATP_Olambda__772,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_wx(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(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_772
tff(fact_10974_ATP_Olambda__773,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_acn((Uu)),(Uua)),(Uub)),Uuc),(Uud)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_acm((Uu)),(Uua)),(Uub)),(Uud)),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_773
tff(fact_10975_ATP_Olambda__774,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_ade(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_add(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_774
tff(fact_10976_ATP_Olambda__775,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_xr(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_xq(Uua),Uuc),Uud)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud)) ).

% ATP.lambda_775
tff(fact_10977_ATP_Olambda__776,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_xp(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_xo(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),Uud)) ).

% ATP.lambda_776
tff(fact_10978_ATP_Olambda__777,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_xn(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_xm(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_777
tff(fact_10979_ATP_Olambda__778,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_xl(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_xk(Uua),Uuc),Uud)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud)) ).

% ATP.lambda_778
tff(fact_10980_ATP_Olambda__779,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_zd(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_779
tff(fact_10981_ATP_Olambda__780,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_zf(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(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_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_780
tff(fact_10982_ATP_Olambda__781,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_ze(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_781
tff(fact_10983_ATP_Olambda__782,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_yl(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_782
tff(fact_10984_ATP_Olambda__783,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_yp(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(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_783
tff(fact_10985_ATP_Olambda__784,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_yn(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_784
tff(fact_10986_ATP_Olambda__785,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_ym(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_785
tff(fact_10987_ATP_Olambda__786,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_yq(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(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(aa_real_fun_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_786
tff(fact_10988_ATP_Olambda__787,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_yo(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_787
tff(fact_10989_ATP_Olambda__788,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_ww(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(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(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uuc)) ).

% ATP.lambda_788
tff(fact_10990_ATP_Olambda__789,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_xc(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_789
tff(fact_10991_ATP_Olambda__790,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_wy(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_790
tff(fact_10992_ATP_Olambda__791,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_xa(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(aa_int_fun_nat_int(power_power_int,Uub),Uud))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_791
tff(fact_10993_ATP_Olambda__792,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_acm((Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_acl((Uu)),(Uua)),(Uub)),(Uuc)),(Uue)),code_bit_cut_integer(Uud)) ).

% ATP.lambda_792
tff(fact_10994_ATP_Olambda__793,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_acl((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_ack((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf)),code_bit_cut_integer(Uue)) ).

% ATP.lambda_793
tff(fact_10995_ATP_Olambda__794,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_afd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(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))),Uud))
           => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I5)),X_12)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I5) ) )
        & $ite(
            Uue = Uuf,
            ! [X3: vEBT_VEBT] :
              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Uua))
             => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X9) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X3)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_794
tff(fact_10996_ATP_Olambda__795,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_acj((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_795
tff(fact_10997_ATP_Olambda__796,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_ack((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_acj((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug)),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_796
tff(fact_10998_ATP_Olambda__797,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_pc,Uu),Uua) = Uu ).

% ATP.lambda_797
tff(fact_10999_ATP_Olambda__798,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pd,Uu),Uua) = Uu ).

% ATP.lambda_798
tff(fact_11000_ATP_Olambda__799,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_nh,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_799
tff(fact_11001_ATP_Olambda__800,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_af,Uu) = Uu ).

% ATP.lambda_800
tff(fact_11002_ATP_Olambda__801,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ag,Uu) = Uu ).

% ATP.lambda_801
tff(fact_11003_ATP_Olambda__802,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_ah,Uu) = Uu ).

% ATP.lambda_802
tff(fact_11004_ATP_Olambda__803,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ai,Uu) = Uu ).

% ATP.lambda_803
tff(fact_11005_ATP_Olambda__804,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aj,Uu) = Uu ).

% ATP.lambda_804
tff(fact_11006_ATP_Olambda__805,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_bi,Uu) = zero_zero_complex ).

% ATP.lambda_805
tff(fact_11007_ATP_Olambda__806,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_bj,Uu) = zero_zero_real ).

% ATP.lambda_806
tff(fact_11008_ATP_Olambda__807,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_bk,Uu) = zero_zero_rat ).

% ATP.lambda_807
tff(fact_11009_ATP_Olambda__808,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_nx,Uu) = zero_zero_complex ).

% ATP.lambda_808
tff(fact_11010_ATP_Olambda__809,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_fb,Uu) = zero_zero_real ).

% ATP.lambda_809
tff(fact_11011_ATP_Olambda__810,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bl,Uu) = zero_zero_nat ).

% ATP.lambda_810
tff(fact_11012_ATP_Olambda__811,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ny,Uu) = zero_zero_int ).

% ATP.lambda_811
tff(fact_11013_ATP_Olambda__812,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_bm,Uu) = zero_zero_int ).

% ATP.lambda_812
tff(fact_11014_ATP_Olambda__813,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_zu,Uu) = one_one_int ).

% ATP.lambda_813
tff(fact_11015_ATP_Olambda__814,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_zv,Uu) = one_one_int ).

% ATP.lambda_814
tff(fact_11016_ATP_Olambda__815,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aw,Uu)
    <=> $false ) ).

% ATP.lambda_815
tff(fact_11017_ATP_Olambda__816,axiom,
    ! [Uu: complex] :
      ( aa_complex_o(aTP_Lamm_ax,Uu)
    <=> $false ) ).

% ATP.lambda_816
tff(fact_11018_ATP_Olambda__817,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_bb,Uu)
    <=> $false ) ).

% ATP.lambda_817
tff(fact_11019_ATP_Olambda__818,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ay,Uu)
    <=> $false ) ).

% ATP.lambda_818
tff(fact_11020_ATP_Olambda__819,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_az,Uu)
    <=> $false ) ).

% ATP.lambda_819
tff(fact_11021_ATP_Olambda__820,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_ba,Uu)
    <=> $false ) ).

% ATP.lambda_820
tff(fact_11022_ATP_Olambda__821,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_abc,Uu)
    <=> $true ) ).

% ATP.lambda_821

% Helper facts (1)
tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X9: real] : aa_real_o(P,X9) ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),
              $ite(
                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx) = ma,
                $let(
                  maxs: option_nat,
                  maxs:= vEBT_vebt_maxt(vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),
                  $ite(maxs = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),the_nat(maxs)))))) ),
                ma ))),deg,list_u1324408373059187874T_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na),vEBT_vebt_delete(nth_VEBT_VEBT(treeList,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na))),vEBT_vebt_delete(summary,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,summin),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))),lx)),na)))),deg) ).

%------------------------------------------------------------------------------